Question

Determine whether the geometric series 27 + 18 + 12 + 8 + ... converges or diverges, and identify the sum if it exists.

Answer 1

Geometric sequence. Sum of a geometric series.

A geometric sequence goes from one term to the next by always multiplying or dividing by the constant value except 0. The constant number multiplied (or divided) at each stage of a geometric sequence is called the common ratio (r).

A geometric series is the sum of an infinite number of terms of a geometric sequence.

A geometric series is convergers if |r| < 1.

A geometric series is diveres if |r| > 1.

Calculate the common ratio:

$r=\dfrac{18}{27}=\dfrac{18:9}{27:9}=\dfrac{2}{3}\\\\r=\dfrac{12}{18}=\dfrac{12:6}{18:6}=\dfrac{2}{3}\\\\r=\dfrac{8}{12}=\dfrac{8:24}{12:4}=\dfrac{2}{3}$

$\left|\dfrac{2}{3}\right| < 1$

The geometric series is converges.

Therefore exist the sum.

Formula of a sum of a geometric series:

$S=\dfrac{a_1}{1-r},\qquad|r| < 1$

Substitute:

$a_1=27,\ r=\dfrac{2}{3}$

$S=\dfrac{27}{1-\frac{2}{3}}=\dfrac{27}{\frac{1}{3}}=27\cdot\dfrac{3}{1}=81$

$\huge\boxed{S=81}$