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
Find the sum of the first $ 12 $ terms of a geometric sequence if $ a_1 = 3 ~~ \text{and} ~~ r = 2 $.
solution
$$ S_{ 12 } = 12285 $$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 ~~\text{and}~~ n = 12 $. After substituting these values into the formula, we obtain:
$$ \begin{aligned} S_n &= a_1 \cdot \frac{1 - r^n}{1-r} \\ S_{ 12 } &= 3 \cdot \frac{1 - 2^{ 12 }}{1- 2 } \\ S_{ 12 } &= 3 \cdot 4095 \\ S_{ 12 } &= 12285 \end{aligned}$$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 $\color{blue}{S_n = a_1 \frac{1-r^n}{1-r}}$.
Please tell me how can I make this better.