Question

How do you do this question?

Answer 1

(-11 + √165) / 22

Explanation:

∑₂°° (1 + c)⁻ⁿ

= ∑₂°° (1 / (1 + c))ⁿ

= ∑₁°° (1 / (1 + c))ⁿ⁺¹

This is a geometric series with a₁ = 1/(1+c)² and r = 1/(1+c).

The sum of an infinite geometric series is:

S = a₁ / (1 − r)

Therefore:

11 = 1/(1+c)² / (1 − 1/(1+c))

11 = 1 / ((1+c)² − (1+c))

11 ((1+c)² − (1+c)) = 1

11 (1 + 2c + c² − 1 − c) = 1

11 (c + c²) = 1

11c² + 11c − 1 = 0

c = [ -11 ± √(11² − 4(11)(-1)) ] / 22

c = (-11 ± √165) / 22

c cannot be between -2 and 0, so c = (-11 + √165) / 22.