Doubling daily wages (geometric sequence): A boy earns ₹1 on day 1, ₹2 on day 2, ₹4 on day 3, doubling each day. If he works from 1st February to 20th February (20 days), how much does he earn in total?

Introduction / Context:
This is a geometric progression (GP) with first term 1 and common ratio 2. The total for n days equals the sum of the first n powers of 2 starting at 2^0. The standard GP sum formula applies directly here.

Given Data / Assumptions:

• Day k earning = 2^(k−1) rupees (k = 1 to 20).
• Number of days = 20 (Feb 1 through Feb 20 inclusive).

Concept / Approach:
Sum of GP: S_n = a * (r^n − 1) / (r − 1). With a = 1 and r = 2, this simplifies to S_n = 2^n − 1. Substitute n = 20 to get the total earnings.

Step-by-Step Solution:

Verification / Alternative check:
Quick check for small n: S_1 = 1 = 2^1 − 1; S_2 = 1 + 2 = 3 = 2^2 − 1. Pattern holds; thus S_20 = 2^20 − 1.

Why Other Options Are Wrong:

• 2^19 − 1: Sum for 19 days, not 20.
• 2^20: Overcounts by 1.
• 2^19: Not a sum; it is a single day’s wage on day 20−1.

Common Pitfalls:
Confusing arithmetic and geometric sequences, or forgetting that the first term is 1 (2^0). Always apply the GP sum formula carefully.

Final Answer: