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Geometric sequences and sums. Infinite geometric series

A sequence of numbers where each term after the first is got by multiplying the previous one by a fixed number. That number is called the common ratio.

For example, the sequence 2, 6, 18, 54, 162, 486, 1458, . . . is a geometric progression with a common ratio of 3.

Rule of geometric sequence

General rule of geometric sequence can be written by this formula, where $b_1$ - the first term, $q$ - the factor between the terms, $n$ - the number of term.

$$b_n=b_1\cdot q^{n-1}$$

Sum of geometric sequence

To sum up the terms of a geometric sequence is simple only when we have not many terms. But for sequences that have large number of terms we can use this formula below.

$$\displaystyle S_n=\frac{b_1(q^n-1)}{q-1}$$

Infinite geometric sequence

If we have geometric sequence, where $-1< q <1$ and number of terms goes to infinity, it is called infinity geometric serie and sum of it can be calculated by formula below.

$$S_n=\frac{b_1}{1-q}$$

2020-12-02

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