How many terms are there in the series 1/1.5 + 1/0.15 + 1/0.015 +... = 74074074 ?

Given:

Series: 1/1.5 + 1/0.15 + 1/0.015 + ... = 74074074

Formula used:

Sum of a Geometric Progression (GP) = a(1 - rⁿ) / (1 - r), 

Calculation:

Given, 2/3 + 20/3 + 200/3 + ... = 74074074

Here, a = 2/3

r = (20/3) / (2/3) = 10

Let 'n' be the number of terms.

Sum = (2/3)(1 - 10ⁿ) / (1 - 10) = 74074074

(2/3)(1 - 10ⁿ) / (-9) = 74074074

(2/3)(10ⁿ - 1) / 9 = 74074074

2(10ⁿ - 1) = 74074074 × 27

10ⁿ - 1 = 74074074 × 27 / 2 = 999999999

10ⁿ = 1000000000

10ⁿ = 10⁹

n = 9

∴ There are 9 terms in the series.