This tool can help you 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.
problem
$$ a_1 = 1 ~,~ r = 5 ~,~ S_n = 3906 ~,~ n = ? $$solution
$$ n = 6 $$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:
$$ \begin{aligned} S_n &= a_1 \cdot \frac{1-r^n}{1-r} \\[1 em] 3906 &= 1 \cdot \frac{ 1-5^n}{1 - 5} \\[1 em] 1-5^n &= \frac{ 3906}{ 1} \cdot \left(1 - 5 \right) \\[1 em] 1-5^n &= -15624 \\[1 em] 5^n &= 15625 \\[1 em] \log \left( 5^n \right) &= \log \left(15625 \right) \\[1 em] n \cdot \log \left( 5 \right) &= \log \left(15625 \right) \\[1 em] n &= 6 \end{aligned} $$The first few terms of this sequence are:
$$ 1, ~~~5, ~~~25, ~~~125 . . . $$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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