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# Geometric sequences calclator

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

$$a_1 = 3 ~~,~~ r = 2 ~~,~~ S_{ 12 } = ?$$

Result

$$S_{ 12 } = 12285$$

Explanation

To find $S_{ 12 }$ we use formula

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

In this example we have $a_1 = 3 ~~,~~ r = 2 ~~,~~ n = 12$. After substituting these values to above formula, we obtain:

\begin{aligned} S_n &= a_1 \cdot \frac{1 - r^n}{1-r} \\ S_{ 12 } &= 3 \cdot \frac{1 - 2^{ 12 }}{1- 2 } \\ S_{ 12 } &= 3 \cdot 4095 \\ S_{ 12 } &= 12285 \end{aligned}

## Report an Error !

Script name : geometric-sequences-calculator

Imputed values: 0 , 3 , 2 , 12 , S

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## Share Result

Geometric Sequences Calculator
0 1 2 3 4 5 6 7 8 9 - / . del
 an ( the the n-th term ) Sn ( the sum of the first n terms )
Show me the solution without steps
Geometric Sequences Calculator
0 1 2 3 4 5 6 7 8 9 - / . del
Show me the solution without steps
Geometric Sequences Calculator
0 1 2 3 4 5 6 7 8 9 / . del
Show me the solution without steps

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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