How can you use the formula for the sum of a finite geometric series to solve problems?

Question

 HALPPPPP PLSSSS SJSII

 

Answer 1

A geometric series has the following property: a_n = r a_(n-1). Similarly, it can be said that a_n / a_(n-1) = r for any pair of numbers. If n = 5, then a₅ / a₄ = r. And if n = 10, a₁₀ /a₉ = r also. For a consecutive series n of numbers whose ratio is r, the sum of the first n terms is precisely

Sum_n = a (1-rⁿ)/(1-r) where r is not allowed to be zero or 1.0 and a is the first number in the sequence

Suppose a = 0.25 and r = 0.7. The first five terms in this series are 0.25,

a₂ = 0.25 x 0.7 = 0.175

a₃ = a₂ x 0.7 = 0.175 x 0.7 = 0.1225

a₄ = a₃ x 0.7 = 0.1225 x 0.7 = 0.08575

a₅ = a₄ x 0.7 = 0.08575 x 0.7 = 0.060025

Summing these five numbers gives 0.693275

No, using the formula stated above, the sum of the first five terms is

Sum₅ = 0.25 x (1-0.7⁵)/(1-0.7) = 0.25 x 0.83193 / 0.3 = 0.693275

The formula gives the same answer as the brute force manual summation.