Q.53 If 1 + r + r² + r³ + … ∞ = 1.5,
then, 1 + 2r + 3r² + 4r³ + … ∞ = (up to two decimal places) ________

Solving Infinite GP Sums — Find the Value of 1 + 2r + 3r² + …

Given:

1 + r + r² + r³ + … = 1.5

Step 1 — Use GP Sum Formula

The formula for an infinite geometric series is:

1 / (1 − r) = 1.5

Solving:

1 − r = 1 / 1.5 = 2/3

r = 1 − 2/3 = 1/3

Step 2 — Apply Formula for 1 + 2r + 3r² + …

The sum of the infinite series:

1 + 2r + 3r² + 4r³ + … = 1 / (1 − r)²

Plug in r = 1/3:

S = 1 / (1 − 1/3)²
= 1 / (2/3)²
= 1 / (4/9)
= 9/4
= 2.25

Final Answer

The sum equals 2.25 (correct to two decimal places).