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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.

Click here to see a list of example problems that can be solved by using this calculator.

**Problem**

**Result**

**Explanation**

To find $ r $ we use formula for the $ n^{th} $ term $\color{blue}{a_n = a_1 \cdot r^{n-1}}$

$$ \begin{aligned} a_{ 4} &= a_1 \cdot r^{ 4 - 1} \\ a_{ 8} &= a_1 \cdot r^{ 8 - 1} \end{aligned}$$To find $ r $ we will divide above equations:

$$ \begin{aligned} \frac{ a_{ 4}}{a_{ 8}} &= \frac { a_1 \cdot r^{ 4 - 1}} { a_1 \cdot r^{ 8 - 1}} \\ \frac{ \frac{ 4 }{ 3 }}{ \frac{ 64 }{ 243 } } &= \frac{ r^{ 3}}{ r^{ 7} } \\ \frac{ 81 }{ 16 } &= \frac{1}{r^{ 4 }} \\ r^{ 4 } &= \frac{ 16 }{ 81 } \\ r &= \sqrt[ 4 ]{ \frac{ 16 }{ 81 } } \\ r &= \frac{ 2 }{ 3 } \end{aligned}$$Share this result with others by using the link below.

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**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 $\color{blue}{S_n = a_1 \frac{1-r^n}{1-r}}$.

**Example problems that can be solved with this calculator **

**Example 1:**Find the sum of series $ \sum\limits_{i=1}^{12} 3\cdot 2^i $

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**Example 2:**
Find the common ratio if the fourth term in geometric series is $\frac{4}{3}$ and the seventh term is $\frac{64}{243}$.

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**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.

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