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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 $ n $ we use formula

$$ \color{blue}{S_n = a_1 \cdot \frac{1-r^n}{1-r}}$$In this example we have $ a_1 = 1 ~~,~~ r = 5 ~~,~~ S_n = 3906 $. After substituting these values to above formula, we obtain:

$$ \begin{aligned} S_n &= a_1 \cdot \frac{1-r^n}{1-r} \\ 3906 &= 1 \cdot \frac{ 1-5^n}{1 - 5} \\ 1-5^n &= \frac{ 3906}{ 1} \cdot \left(1 - 5 \right) \\ 1-5^n &= -15624 \\ 5^n &= 15625 \\ \log \left( 5^n \right) &= \log \left(15625 \right) \\ n \cdot \log \left( 5 \right) &= \log \left(15625 \right) \\ n &= 6 \end{aligned}$$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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