How to calculate the value of i=?

0.200

Step-by-step explanation:

∑ₓ₌₁⁴⁰ (1 + i)⁻ˣ = 5

∑ₓ₌₁⁴⁰ ((1 + i)⁻¹)ˣ = 5

This is a geometric series.  The sum of the first n terms of a geometric series is:

S = a₁ (1 − rⁿ) / (1 − r)

where a₁ is the first term and r is the common ratio.

Here, a₁ = (1 + i)⁻¹, r = (1 + i)⁻¹, and n = 40.  For simplicity, let's write both a₁ and r in terms of r.  So a₁ = r and r = r.

5 = r (1 − r⁴⁰) / (1 − r)

5 (1 − r) = r (1 − r⁴⁰)

5 − 5r = r − r⁴¹

r⁴¹ − 6r + 5 = 0

Solving with a calculator, r ≈ 0.8334.

Therefore:

(1 + i)⁻¹ = 0.8334

1 + i = 1.200

i = 0.200

answer: x=60