maths.free › Precalculus › Sequences and series
Sequences and series
Arithmetic and geometric sequences, partial sums, and the geometric series formula.
An arithmetic sequence adds the same amount each step; a geometric one multiplies by the same ratio. Their sums have closed forms, n(a₁ + aₙ)/2 and a(rⁿ − 1)/(r − 1), and when |r| < 1 the geometric series keeps converging as n grows, which is the first infinite sum most people meet.
مثال کار شده · sum of 2^k for k = 0 to 10
قدم به قدم
- \sum_{k=0}^{10} 2^{k}
Write the sum out.
- = 2047
Closed form.
جواب رو نشون بده
حالا تو يه مساله انتخاب کن، يا خودت رو بنويس يا بکشي هر گام، يه تصوير، جواب مخفي ميشه تا وقتي که تو بپرسي
نمادهای استفاده‌شده در اینجا
هر نماد را برای تعریف کامل، تصویر و معنی هر حرف در آن بزنید.
چطور: Sequences and series
- Write the sum out.
- Closed form.
سوالاتي که مردم ميپرسن
What is a function, really?
A rule that assigns exactly one output to each input. The vertical-line test on a graph is the same idea: no input may have two outputs.
Why do we need complex numbers?
Because x² + 1 = 0 has no real solution, and allowing one new number i with i² = −1 makes every polynomial equation solvable. They then turn out to describe rotation, waves and alternating current more naturally than real numbers do.
بیشتر در Precalculus
Complex numbersPolynomial functionsRational functionsThe binomial theoremConic sectionsVectorsExponential and logarithmic functionsPolynomial division and the remainder theoremParametric equations and polar coordinates