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- Geometric Sequences Calculator

This tool can help you to find $n^{th}$ term and the sum of the first $n$ terms of a geometric progression. Also, this calculator can be used to solve more complicated problems. For example, the calculator can find the first term ($a_1$) and common ratio ($r$) if $a_2 = 6 $ and $a_5 = 48$. The calculator will generate all the work with detailed explanation.

examples

example 1:

Find the sum $\sum\limits_{i=0}^{12} 3 \cdot 2^i $.

example 2:

Find the common ratio if the fourth term in geometric series is $\frac{4}{3}$ and the eighth term is $\frac{64}{243}$.

example 3:

The first term of an geometric progression is 1, and the common ratio is 5 determine how many terms must be added
together to give a sum of 3906.

**Definition:**

Geometric sequence is a list of numbers where each term is obtained by multiplying the previous term by a constant. The constant is called the common ratio ($r$).

**Formulas:**

The formula for finding $n^{th}$ term of a geometric progression is $\color{blue}{a_n = a_1 \cdot r^{n-1}}$, where $\color{blue}{a_1}$ is the first term and $\color{blue}{r}$ is the common ratio. The formulas for the sum of first $n$ numbers are $\color{blue}{S_n = a_1 \frac{1-r^n}{1-r}}$.

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