Geometric sequence

Question

The sum of the first 5 terms of a geometric sequence is 93/4 and the first term is 12. What is the common ratio?

Answer 1

Working off of Mark H.'s cogent presentation

93/48 = (r⁵ - 1) / (r - 1)

93/48 = x⁴ + x³ + x² + x - 1

0 = x⁴ + x³ + x² + x - 45/48

0 = x⁴ + x³ + x² + x - 15/16

By Descartes Rule of Signs at most one positive real root.

By Rational Root Theorem root must be a factor of 15/16.

Since the common ratio is to the fourth power and 0.5⁴ = 1/16 I am going to try 0.5.

And like Mark H.'s calculation, it works

Answer 2

The formula for the sum of the terms:

Sum = a(1) * ( (1 - r^n) / ( 1 - r) )

n = # of terms

r = common ratio

a(1) = first term

Write the equation:

93/4 = 12 * ( (1 - r^5) / (1 - r) )

93/48 = (r^5 - 1) / (r - 1)

From a graphing tool, r = 0.5, but I'm not clear on how to get that from the formula.

To verify: 93/48 = (0.5^5 - 1)/(0.5 - 1) = (31/32)/0.5 = 31/16 = 93/48

Answer 3

There are two real answers and two complex answers to this problem. To see this,

let r be the common ratio. If the first term is 12, then the sum of the first 5 terms is:

12 + 12t + 12r² + 12r³ + 12r⁴ = 93 / 4

Below is a graph of the equation y(r) = (12 - 93/4) + 12t + 12r² + 12r³ + 12r⁴

https://www.desmos.com/calculator/novheb063x

The real roots are r = 1/2 and -1.27839

The complex roots are r = -0.11081 +/- 1.20599i

The problem statement does not explicitly state that the common ratio be real, so we have to include the complex cases. At the Pre-calculus level, only real roots are usually considered.