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Arithmetic Sequences

Determine if a sequence is arithmetic

Determine if a Sequence is Arithmetic

The last section introduced sequences and now we will look at two specific types of sequences that each have special properties. In this section we will look at arithmetic sequences and in the next section, geometric sequences.

An arithmetic sequence is a sequence where the difference between consecutive terms is constant. The difference between consecutive terms in an arithmetic sequence, \({a}_{n}-{a}_{n-1},\) is d, the common difference, for n greater than or equal to two.

In each of these sequences, the difference between consecutive terms is constant, and so the sequence is arithmetic.

Example

Try it.

Determine if each sequence is arithmetic. If so, indicate the common difference.

ⓐ \(5,9,13,17,21,25\text{,}\ \text{\ldots }\)

ⓑ \(4,9,12,17,20,25\text{,}\ \text{\ldots }\)

Solution

To determine if the sequence is arithmetic, we find the difference of the consecutive terms shown.

ⓐ

Find the difference of the consecutive terms.	\(\begin{array}{lllllllllll}5, & & \ 9, & & \ 13, & & \ 17 & & \ 21, & \ 25,\ldots \\ \\ \\ 9-5 & & \ 13-9 & & \ 17-13 & & \ 21-17 & & \ 25-21 & & \\ 4 & & \ 4 & & \ 4 & & \ 4 & & \ 4 & & \end{array}\)
	The sequence is arithmetic. The common difference is \(d=4\text{.}\)

ⓑ

Find the difference of the consecutive terms.	\(\begin{array}{lllllllllll}4, & & \ 9, & & \ 12, & & \ 17 & & \ 20, & \ 25,\ldots \\ \\ \\ 9-4 & & \ 12-9 & & \ 17-12 & & \ 20-17 & & \ 25-20 & & \\ 5 & & \ 3 & & \ 5 & & \ 3 & & \ 5 & & \end{array}\)
	The sequence is not arithmetic as all the differences between the consecutive terms are not the same. There is no common difference.

ⓒ

Find the difference of the consecutive terms.	\(\begin{array}{lllllllllll}10, & & \ 3, & & \ -4, & & \ -11 & & \ -18, & \ -25,\ldots & \\ \\ \\ 3-10 & & \ -4-3 & & \ -11-(-4) & & \ -18-(-11) & & \ -25-(-18) & & \\ -7 & & \ -7 & & \ -7 & & \ -7 & & \ -7 & & \end{array}\)
	The sequence is arithmetic. The common difference is \(d=-7\text{.}\)

If we know the first term, \({a}_{1},\) and the common difference, d, we can list a finite number of terms of the sequence.

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Find the General Term (

Just as we found a formula for the general term of a sequence, we can also find a formula for the general term of an arithmetic sequence.

Let’s write the first few terms of a sequence where the first term is \({a}_{1}\) and the common difference is d. We will then look for a pattern.

As we look for a pattern we see that each term starts with \({a}_{1}\).

The first term adds 0d to the \({a}_{1}\), the second term adds 1d, the third term adds 2d, the fourth term adds 3d, and the fifth term adds 4d. The number of ds that were added to \({a}_{1}\) is one less than the number of the term. This leads us to the following

\[{a}_{n}={a}_{1}+(n-1)d\]

We will use this formula in the next example to find the 15^th term of a sequence.

Example

Try it.

Find the fifteenth term of a sequence where the first term is 3 and the common difference is 6.

Solution

To find the fifteenth term, \({a}_{15},\) use the formula with \({a}_{1}=3\ \text{and}\ d=6\text{.}\)	\(\ {a}_{n}={a}_{1}+(n-1)d\)
Substitute in the values.	\(\ {a}_{15}=3+(15-1)6\)
Simplify.	\(\ {a}_{15}=3+(14)6\)
	\(\ {a}_{15}=87\)

Sometimes we do not know the first term and we must use other given information to find it before we find the requested term.

Example

Try it.

Find the twelfth term of a sequence where the seventh term is 10 and the common difference is \(-2\). Give the formula for the general term.

Solution

To first find the first term, \({a}_{1},\) use the formula with \({a}_{7}=10,n=7,\ \text{and}\ d=-2.\)	\(\ {a}_{n}={a}_{1}+(n-1)d\)
Substitute in the values.	\(\ 10={a}_{1}+(7-1)(-2)\)
Simplify.	\(\ 10={a}_{1}+(6)(-2)\)
	\(\ 10={a}_{1}-12\)
	\(\ {a}_{1}=22\)
Find the twelfth term, \({a}_{12},\) using the formula with \({a}_{1}=22,n=12,\ \text{and}\ d=-2\text{.}\)	\(\ {a}_{n}={a}_{1}+(n-1)d\)
Substitute in the values.	\(\ {a}_{12}=22+(12-1)(-2)\)
Simplify.	\(\ {a}_{12}=22+(11)(-2)\)
	\(\ {a}_{12}=0\)
	\(\\)The twelfth term of the sequence is 0, \({a}_{12}=0.\)
To find the general term, substitute the values into the formula.	\(\ {a}_{n}={a}_{1}+(n-1)d\)
	\(\ {a}_{n}=22+(n-1)(-2)\)
	\(\ {a}_{n}=22-2n+2\)
	\(\\)The general term is \({a}_{n}=-2n+24.\)

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Find the Sum of the First

As with the general sequences, it is often useful to find the sum of an arithmetic sequence. The sum, \({S}_{n},\) of the first \(n\) terms of any arithmetic sequence is written as \({S}_{n}={a}_{1}+{a}_{2}+{a}_{3}+...+{a}_{n}.\) To find the sum by merely adding all the terms can be tedious. So we can also develop a formula to find the sum of a sequence using the first and last term of the sequence.

We can develop this new formula by first writing the sum by starting with the first term, \({a}_{1},\) and keep adding a d to get the next term as:

\[{S}_{n}={a}_{1}+({a}_{1}+d)+({a}_{1}+2d)+\ldots +{a}_{n}.\]

We can also reverse the order of the terms and write the sum by starting with \({a}_{n}\) and keep subtracting d to get the next term as

\[{S}_{n}={a}_{n}+({a}_{n}-d)+({a}_{n}-2d)+\ldots +{a}_{1}.\]

If we add these two expressions for the sum of the first n terms of an arithmetic sequence, we can derive a formula for the sum of the first n terms of any arithmetic series.

\[\begin{array}{lllllllllllllllllllll}\underset{\text{_________________________________________________________}}{\begin{array}{lllllllllll}{S}_{n} & = & {a}_{1} & \ + & ({a}_{1}+d) & + & ({a}_{1}+2d) & + & \ldots & + & {a}_{n} \\ +{S}_{n} & = & {a}_{n} & \ + & ({a}_{n}-d) & + & ({a}_{n}-2d) & + & \ldots & + & {a}_{1}\end{array}} \\ \\ 2{S}_{n}\ =\ ({a}_{1}+{a}_{n})+\ ({a}_{1}+{a}_{n})\ +\ ({a}_{1}+{a}_{n})\ +\ \ldots \ +\ ({a}_{1}+{a}_{n})\end{array}\]

Because there are n sums of \(({a}_{1}+{a}_{n})\) on the right side of the equation, we rewrite the right side as \(n({a}_{1}+{a}_{n}).\)

\[2{S}_{n}=n({a}_{1}+{a}_{n})\]

We divide by two to solve for \({S}_{n}.\)

\[{S}_{n}=\frac{n}{2}({a}_{1}+{a}_{n})\]

This gives us a general formula for the sum of the first n terms of an arithmetic sequence.

Example

Try it.

Find the sum of the first 30 terms of the arithmetic sequence: 8, 13, 18, 23, 28, …

Solution

To find the sum, we will use the formula \({S}_{n}=\frac{n}{2}({a}_{1}+{a}_{n}).\) We know \({a}_{1}=8\), \(d=5\) and \(n=30,\) but we need to find \({a}_{n}\) in order to use the sum formula.

	\(\ {a}_{n}={a}_{1}+(n-1)d\)
\(\text{Find}\ {a}_{n}\ \text{where}\ {a}_{1}=8,d=5\ \text{and}\ n=30.\)	\({a}_{30}=8+(30-1)5\)
Simplify.	\({a}_{30}=8+(29)5\)
	\({a}_{30}=153\)
\(\text{Knowing}\ {a}_{1}=8,n=30,\ \text{and}\ {a}_{30}=153,\) use the sum formula.	\(\ {S}_{n}=\frac{n}{2}({a}_{1}+{a}_{n})\)
Substitute in the values.	\({S}_{30}=\frac{30}{2}(8+153)\)
Simplify.	\({S}_{30}=15(161)\)
Simplify.	\({S}_{30}=2,415\)

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Key Concepts

General Term (nth term) of an Arithmetic Sequence
The general term of an arithmetic sequence with first term \({a}_{1}\) and the common difference d is
\[{a}_{n}={a}_{1}+(n-1)d\]
Sum of the First n Terms of an Arithmetic Sequence
The sum, \({S}_{n},\) of the first n terms of an arithmetic sequence, where \({a}_{1}\) is the first term and \({a}_{n}\) is the nth term is
\[{S}_{n}=\frac{n}{2}({a}_{1}+{a}_{n})\]

Arithmetic Sequences

Determine if a Sequence is Arithmetic

In the following exercises, determine if each sequence is arithmetic, and if so, indicate the common difference.

Try it.

\(4,12,20,28,36,44\text{,}\ \text{\ldots }\)

Solution

The sequence is arithmetic with common difference \(d=8.\)

Try it.

\(-7,-2,3,8,13,18\text{,}\ \text{\ldots }\)

Try it.

\(-15,-16,3,12,21,30\text{,}\ \text{\ldots }\)

Solution

The sequence is not arithmetic.

Try it.

\(11,5,-1,-7-13,-19\text{,}\ \text{\ldots }\)

Try it.

\(8,5,2,-1,-4,-7\text{,}\ \text{\ldots }\)

Solution

The sequence is arithmetic with common difference \(d=-3.\)

Try it.

\(15,5,-5,-15,-25,-35\text{,}\ \text{\ldots }\)

In the following exercises, write the first five terms of each sequence with the given first term and common difference.

Try it.

\({a}_{1}=11\) and \(d=7\)

Solution

\(11,18,25,32,39\)

Try it.

\({a}_{1}=18\) and \(d=9\)

Try it.

\({a}_{1}=-7\) and \(d=4\)

Solution

\(-7,-3,1,5,9\)

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\({a}_{1}=-8\) and \(d=5\)

Try it.

\({a}_{1}=14\) and \(d=-9\)

Solution

\(14,5,-4,-13,-22\)

Try it.

\({a}_{1}=-3\) and \(d=-3\)

Find the General Term (nth Term) of an Arithmetic Sequence

In the following exercises, find the term described using the information provided.

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Find the twenty-first term of a sequence where the first term is three and the common difference is eight.

Solution

\(163\)

Try it.

Find the twenty-third term of a sequence where the first term is six and the common difference is four.

Try it.

Find the thirtieth term of a sequence where the first term is \(-14\) and the common difference is five.

Solution

\(131\)

Try it.

Find the fortieth term of a sequence where the first term is \(-19\) and the common difference is seven.

Try it.

Find the sixteenth term of a sequence where the first term is 11 and the common difference is \(-6.\)

Solution

\(-79\)

Try it.

Find the fourteenth term of a sequence where the first term is eight and the common difference is \(-3.\)

Try it.

Find the twentieth term of a sequence where the fifth term is \(-4\) and the common difference is \(-2.\) Give the formula for the general term.

Solution

\({a}_{20}=-34.\) The general term is \({a}_{n}=-2n+6.\)

Try it.

Find the thirteenth term of a sequence where the sixth term is \(-1\) and the common difference is \(-4.\) Give the formula for the general term.

Try it.

Find the eleventh term of a sequence where the third term is 19 and the common difference is five. Give the formula for the general term.

Solution

\({a}_{11}=59.\) The general term is \({a}_{n}=5n+4.\)

Try it.

Find the fifteenth term of a sequence where the tenth term is 17 and the common difference is seven. Give the formula for the general term.

Try it.

Find the eighth term of a sequence where the seventh term is \(-8\) and the common difference is \(-5.\) Give the formula for the general term.

Solution

\({a}_{8}=-13.\) The general term is \({a}_{n}=-5n+27.\)

Try it.

Find the fifteenth term of a sequence where the tenth term is \(-11\) and the common difference is \(-3.\) Give the formula for the general term.

In the following exercises, find the first term and common difference of the sequence with the given terms. Give the formula for the general term.

Try it.

The second term is 14 and the thirteenth term is 47.

Solution

\({a}_{1}=11,\)\(d=3.\) The general term is \({a}_{n}=3n+8.\)

Try it.

The third term is 18 and the fourteenth term is 73.

Try it.

The second term is 13 and the tenth term is \(-51.\)

Solution

\({a}_{1}=21,\)\(d=-8.\) The general term is \({a}_{n}=-8n+29.\)

Try it.

The third term is four and the tenth term is \(-38\).

Try it.

The fourth term is \(-6\) and the fifteenth term is 27.

Solution

\({a}_{1}=-15,\)\(d=3.\) The general term is \({a}_{n}=3n-18.\)

Try it.

The third term is \(-13\) and the seventeenth term is 15.

Find the Sum of the First n Terms of an Arithmetic Sequence

In the following exercises, find the sum of the first 30 terms of each arithmetic sequence.

Try it.

\(11,14,17,20,23\text{,}\ \text{\ldots }\)

Solution

1,635

Try it.

\(12,18,24,30,36\text{,}\ \text{\ldots }\)

Try it.

\(8,5,2,-1,-4\text{,}\ \text{\ldots }\)

Solution

\(-1,065\)

Try it.

\(16,10,4,-2,-8\text{,}\ \text{\ldots }\)

Try it.

\(-17,-15,-13,-11,-9\text{,}\ \text{\ldots }\)

Solution

360

Try it.

\(-15,-12,-9,-6,-3\text{,}\ \text{\ldots }\)

In the following exercises, find the sum of the first 50 terms of the arithmetic sequence whose general term is given.

Try it.

\({a}_{n}=5n-1\)

Solution

6,325

Try it.

\({a}_{n}=2n+7\)

Try it.

\({a}_{n}=-3n+5\)

Solution

–3,575

Try it.

\({a}_{n}=-4n+3\)

In the following exercises, find each sum.

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\(\sum _{i=1}^{40}(8i-7)\)

Solution

6,280

Try it.

\(\sum _{i=1}^{45}(7i-5)\)

Try it.

\(\sum _{i=1}^{50}(3i+6)\)

Solution

4,125

Try it.

\(\sum _{i=1}^{25}(4i+3)\)

Try it.

\(\sum _{i=1}^{35}(-6i-2)\)

Solution

\(-3,850\)

Try it.

\(\sum _{i=1}^{30}(-5i+1)\)

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Evaluate \(4n-1\) for the integers 1, 2, 3, and 4.
If you missed this problem, review .

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3, 7, 11, 15
Solve the system of equations: \(\{\begin{array}{l}x+y=7 \\ 3x+4y=23\end{array}.\)
If you missed this problem, review .

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\((5,2)\)
If \(f(n)=\frac{n}{2}(3n+5),\) find \(f(1)+f(20).\)
If you missed this problem, review .

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654

Determine if each sequence is arithmetic. If so, indicate the common difference.

ⓐ \(5,9,13,17,21,25\text{,}\ \text{\ldots }\)

ⓑ \(4,9,12,17,20,25\text{,}\ \text{\ldots }\)

Откройте ответ.

To determine if the sequence is arithmetic, we find the difference of the consecutive terms shown.

ⓐ

Find the difference of the consecutive terms.	\(\begin{array}{lllllllllll}5, & & \ 9, & & \ 13, & & \ 17 & & \ 21, & \ 25,\ldots \\ \\ \\ 9-5 & & \ 13-9 & & \ 17-13 & & \ 21-17 & & \ 25-21 & & \\ 4 & & \ 4 & & \ 4 & & \ 4 & & \ 4 & & \end{array}\)
	The sequence is arithmetic. The common difference is \(d=4\text{.}\)

ⓑ

Find the difference of the consecutive terms.	\(\begin{array}{lllllllllll}4, & & \ 9, & & \ 12, & & \ 17 & & \ 20, & \ 25,\ldots \\ \\ \\ 9-4 & & \ 12-9 & & \ 17-12 & & \ 20-17 & & \ 25-20 & & \\ 5 & & \ 3 & & \ 5 & & \ 3 & & \ 5 & & \end{array}\)
	The sequence is not arithmetic as all the differences between the consecutive terms are not the same. There is no common difference.

ⓒ

Find the difference of the consecutive terms.	\(\begin{array}{lllllllllll}10, & & \ 3, & & \ -4, & & \ -11 & & \ -18, & \ -25,\ldots & \\ \\ \\ 3-10 & & \ -4-3 & & \ -11-(-4) & & \ -18-(-11) & & \ -25-(-18) & & \\ -7 & & \ -7 & & \ -7 & & \ -7 & & \ -7 & & \end{array}\)
	The sequence is arithmetic. The common difference is \(d=-7\text{.}\)

Determine if each sequence is arithmetic. If so, indicate the common difference.
ⓐ \(9,20,31,42,53,64\text{,}\ \text{\ldots }\) ⓑ \(12,6,0,-6,-12,-18\text{,}\ \text{\ldots }\) ⓒ \(7,1,10,4,13,7\text{,}\ \text{\ldots }\)

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ⓐ The sequence is arithmetic with common difference \(d=11\). ⓑ The sequence is arithmetic with common difference \(d=-6\).
ⓒ The sequence is not arithmetic as all the differences between the consecutive terms are not the same.
Determine if each sequence is arithmetic. If so, indicate the common difference.
ⓐ \(-4,4,2,10,8,16\text{,}\ \text{\ldots }\) ⓑ \(-3,-1,1,3,5,7\text{,}\ \text{\ldots }\) ⓒ \(7,2,-3,-8,-13,-18\text{,}\ \text{\ldots }\)

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ⓐ The sequence is not arithmetic as all the differences between the consecutive terms are not the same. ⓑ The sequence is arithmetic with common difference \(d=2.\)
ⓒ The sequence is arithmetic with common difference \(d=-5.\)
Write the first five terms of the sequence where the first term is 5 and the common difference is \(d=-6.\)

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We start with the first term and add the common difference. Then we add the common difference to that result to get the next term, and so on.
\[\begin{array}{lllllllllllllll}{a}_{1} & & & {a}_{2} & & & {a}_{3} & & & {a}_{4} & & & {a}_{5} & & \\ 5 & & & 5+(-6) & & & -1+(-6) & & & -7+(-6) & & & -13+(-6) \\ & & & -1 & & & -7 & & & -13 & & & -19\end{array}\]
The sequence is \(5,-1,-7,-13,-19\text{,}\ \text{\ldots }\)
Write the first five terms of the sequence where the first term is 7 and the common difference is \(d=-4.\)

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\(7,3,-1,-5,-9\text{,}\ \text{\ldots }\)
Write the first five terms of the sequence where the first term is 11 and the common difference is \(d=-8.\)

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\(11,3,-5,-13,-21\text{,}\ \text{\ldots }\)

Find the fifteenth term of a sequence where the first term is 3 and the common difference is 6.

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To find the fifteenth term, \({a}_{15},\) use the formula with \({a}_{1}=3\ \text{and}\ d=6\text{.}\)	\(\ {a}_{n}={a}_{1}+(n-1)d\)
Substitute in the values.	\(\ {a}_{15}=3+(15-1)6\)
Simplify.	\(\ {a}_{15}=3+(14)6\)
	\(\ {a}_{15}=87\)

Find the twenty-seventh term of a sequence where the first term is 7 and the common difference is 9.

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241
Find the eighteenth term of a sequence where the first term is 13 and the common difference is \(-7\).

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\(-106\)

Find the twelfth term of a sequence where the seventh term is 10 and the common difference is \(-2\). Give the formula for the general term.

Откройте ответ.

To first find the first term, \({a}_{1},\) use the formula with \({a}_{7}=10,n=7,\ \text{and}\ d=-2.\)	\(\ {a}_{n}={a}_{1}+(n-1)d\)
Substitute in the values.	\(\ 10={a}_{1}+(7-1)(-2)\)
Simplify.	\(\ 10={a}_{1}+(6)(-2)\)
	\(\ 10={a}_{1}-12\)
	\(\ {a}_{1}=22\)
Find the twelfth term, \({a}_{12},\) using the formula with \({a}_{1}=22,n=12,\ \text{and}\ d=-2\text{.}\)	\(\ {a}_{n}={a}_{1}+(n-1)d\)
Substitute in the values.	\(\ {a}_{12}=22+(12-1)(-2)\)
Simplify.	\(\ {a}_{12}=22+(11)(-2)\)
	\(\ {a}_{12}=0\)
	\(\\)The twelfth term of the sequence is 0, \({a}_{12}=0.\)
To find the general term, substitute the values into the formula.	\(\ {a}_{n}={a}_{1}+(n-1)d\)
	\(\ {a}_{n}=22+(n-1)(-2)\)
	\(\ {a}_{n}=22-2n+2\)
	\(\\)The general term is \({a}_{n}=-2n+24.\)

Find the eleventh term of a sequence where the ninth term is 8 and the common difference is \(-3.\) Give the formula for the general term.

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\({a}_{11}=2.\) The general term is \({a}_{n}=-3n+35.\)
Find the nineteenth term of a sequence where the fifth term is 1 and the common difference is \(-4.\) Give the formula for the general term.

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\({a}_{19}=-55.\) The general term is \({a}_{n}=-4n+21.\)

Find the first term and common difference of a sequence where the fifth term is 19 and the eleventh term is 37. Give the formula for the general term.

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Since we know two terms, we can make a system of equations using the formula for the general term.


We know the value of \({a}_{5}\) and \({a}_{11},\) so we will use \(n=5\) and \(n=11.\)
Substitute in the values, \({a}_{5}=19\) and \({a}_{11}=37.\)
Simplify.
Prepare to eliminate the \({a}_{1}\) term by multiplying the top equation by \(-1.\) Add the equations.
Substituting \(d=3\) back into the first equation.
Solve for \({a}_{1}.\)
Use the formula with \({a}_{1}=7\) and \(d=3.\)
Substitute in the values.
Simplify.	\(\\)
	The first term is \({a}_{1}=7.\) The common difference is \(d=3.\)
	The general term of the sequence is \({a}_{n}=3n+4.\)

Find the first term and common difference of a sequence where the fourth term is 17 and the thirteenth term is 53. Give the formula for the general term.

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\({a}_{1}=5,\)\(d=4.\) The general term is \({a}_{n}=4n+1.\)
Find the first term and common difference of a sequence where the third term is 2 and the twelfth term is \(-25.\) Give the formula for the general term.

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\({a}_{1}=8,\)\(d=-3.\) The general term is \({a}_{n}=-3n+11.\)

Find the sum of the first 30 terms of the arithmetic sequence: 8, 13, 18, 23, 28, …

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To find the sum, we will use the formula \({S}_{n}=\frac{n}{2}({a}_{1}+{a}_{n}).\) We know \({a}_{1}=8\), \(d=5\) and \(n=30,\) but we need to find \({a}_{n}\) in order to use the sum formula.

	\(\ {a}_{n}={a}_{1}+(n-1)d\)
\(\text{Find}\ {a}_{n}\ \text{where}\ {a}_{1}=8,d=5\ \text{and}\ n=30.\)	\({a}_{30}=8+(30-1)5\)
Simplify.	\({a}_{30}=8+(29)5\)
	\({a}_{30}=153\)
\(\text{Knowing}\ {a}_{1}=8,n=30,\ \text{and}\ {a}_{30}=153,\) use the sum formula.	\(\ {S}_{n}=\frac{n}{2}({a}_{1}+{a}_{n})\)
Substitute in the values.	\({S}_{30}=\frac{30}{2}(8+153)\)
Simplify.	\({S}_{30}=15(161)\)
Simplify.	\({S}_{30}=2,415\)

Find the sum of the first 30 terms of the arithmetic sequence: 5, 9, 13, 17, 21, …

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1,890
Find the sum of the first 30 terms of the arithmetic sequence: 7, 10, 13, 16, 19, …

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1,515

Find the sum of the first 50 terms of the arithmetic sequence whose general term is \({a}_{n}=3n-4.\)

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To find the sum, we will use the formula \({S}_{n}=\frac{n}{2}({a}_{1}+{a}_{n}).\) We know \(n=50,\) but we need to find \({a}_{1}\) and \({a}_{n}\) in order to use the sum formula.


Find \({a}_{1},\) by substituting \(n=1.\)
Find \({a}_{n}\) by substituting \(n=50.\)
Simplify.
Knowing \(n=50,\)\({a}_{1}=-1,\) and \({a}_{50}=146\) use the sum formula.
Substitute in the values.
Simplify.
Simplify.

Find the sum of the first 50 terms of the arithmetic sequence whose general term is \({a}_{n}=2n-5.\)

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2,300
Find the sum of the first 50 terms of the arithmetic sequence whose general term is \({a}_{n}=4n+3.\)

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5,250

Find the sum: \(\sum _{i=1}^{25}(4i+7).\)

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To find the sum, we will use the formula \({S}_{n}=\frac{n}{2}({a}_{1}+{a}_{n}).\) We know \(n=25,\) but we need to find \({a}_{1}\) and \({a}_{n}\) in order to use the sum formula.

Expand the summation notation.
Simplify.
Identify \({a}_{1}.\)
Identify \({a}_{25}.\)
Knowing \(n=25,\)\({a}_{1}=11,\) and \({a}_{25}=107\) use the sum formula.
Substitute in the values.
Simplify.
Simplify.

Find the sum: \(\sum _{i=1}^{30}(6i-4).\)

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2,670
Find the sum: \(\sum _{i=1}^{35}(5i-3).\)

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3,045
\(4,12,20,28,36,44\text{,}\ \text{\ldots }\)

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The sequence is arithmetic with common difference \(d=8.\)
\(-7,-2,3,8,13,18\text{,}\ \text{\ldots }\)
\(-15,-16,3,12,21,30\text{,}\ \text{\ldots }\)

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The sequence is not arithmetic.
\(11,5,-1,-7-13,-19\text{,}\ \text{\ldots }\)
\(8,5,2,-1,-4,-7\text{,}\ \text{\ldots }\)

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The sequence is arithmetic with common difference \(d=-3.\)
\(15,5,-5,-15,-25,-35\text{,}\ \text{\ldots }\)
\({a}_{1}=11\) and \(d=7\)

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\(11,18,25,32,39\)
\({a}_{1}=18\) and \(d=9\)
\({a}_{1}=-7\) and \(d=4\)

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\(-7,-3,1,5,9\)
\({a}_{1}=-8\) and \(d=5\)
\({a}_{1}=14\) and \(d=-9\)

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\(14,5,-4,-13,-22\)
\({a}_{1}=-3\) and \(d=-3\)
Find the twenty-first term of a sequence where the first term is three and the common difference is eight.

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\(163\)

Symbols used here

\sum_{k=1}^{n} a_k

summation
Add a_k for k = 1 up to n.

imaginary unit
i² = −1.

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\neq

not equal
The two sides are different.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

|x|

absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.

\log_b x,\ \ln x

logarithm, natural log
The exponent b must be raised to for x; ln uses base e.

\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}

number sets
Naturals, integers, rationals, reals, complex numbers.

How to: Arithmetic Sequences

Determine if a sequence is arithmetic
Find the general term (
Find the sum of the first

Questions people ask

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

Попробуй сам.

Parts of this page are adapted from OpenStax Intermediate Algebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

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