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Geometric Sequence Calculator
Solve geometric progressions step by step
The calculator will find the terms, common ratio, sum of the first $$$n$$$ terms and, if possible, the infinite sum of the geometric sequence from the given data, with steps shown.
Related calculator: Arithmetic Sequence Calculator
Your Input
Find $$$a_{n}$$$, $$$a_{1,2,3,4,5}$$$, $$$a_{4}$$$, $$$S_{3}$$$, $$$S_{\infty}$$$, given $$$a_{1} = 3$$$, $$$r = 5$$$.
Solution
We have that $$$a_{1} = 3$$$.
We have that $$$r = 5$$$.
The formula is $$$a_{n} = a_{1} r^{n - 1} = 3 \cdot 5^{n - 1} = \frac{3 \cdot 5^{n}}{5}$$$.
The first five terms are $$$3$$$, $$$15$$$, $$$75$$$, $$$375$$$, $$$1875$$$.
$$$a_{4} = a_{1} r^{4 - 1} = 3 \cdot 5^{4 - 1} = 375$$$
$$$S_{3} = \frac{a_{1} \left(1 - r^{3}\right)}{1 - r} = \frac{3 \left(1 - 5^{3}\right)}{1 - 5} = 93$$$
Since $$$\left|{r}\right| = 5 \geq 1$$$, the infinite sum is infinite.
Answer
The formula is $$$a_{n} = \frac{3 \cdot 5^{n}}{5} = 0.6 \cdot 5^{n}$$$A.
The first five terms are $$$a_{1,2,3,4,5} = 3, 15, 75, 375, 1875$$$A.
$$$a_{4} = 375$$$A
$$$S_{3} = 93$$$A