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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

$$a_{ 4 } = \frac{ 4 }{ 3 } ~~,~~ a_{ 8 } = \frac{ 64 }{ 243 } ~~,~~ a_1 = ? ~~,~~ r = ?$$

Result

$$r = \frac{ 2 }{ 3 }$$

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}

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Script name : geometric-sequences-calculator

Imputed values: 1 , 4 , 4/3 , 8 , 64/243

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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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