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=(18)/(27)=(18:9)/(27:9)=(2)/(3)\\\\r=(12)/(18)=(12:6)/(18:6)=(2)/(3)\\\\r=(8)/(12)=(8:24)/(12:4)=(2)/(3)`

`\left|(2)/(3)\right| < 1`

The geometric series is converges.

Therefore exist the sum.

Formula of a sum of a geometric series:

`S=(a_1)/(1-r),\qquad|r| < 1`

Substitute:

`a_1=27,\ r=(2)/(3)`

`S=(27)/(1-(2)/(3))=(27)/((1)/(3))=27\cdot(3)/(1)=81`

`\huge\boxed{S=81}`