Geometric earnings: ₹5 on day 1, ₹15 on day 2, ₹45 on day 3, and so on (ratio 3). How much does the boy earn in total by the end of 7 days?

Introduction / Context:
The daily earnings form a geometric progression (GP) with first term a = 5 and common ratio r = 3. The total by day 7 is the sum of the first 7 GP terms.

Given Data / Assumptions:

• a = 5 (₹), r = 3.
• Number of days n = 7.

Concept / Approach:
Use the GP sum formula for r ≠ 1: S_n = a * (r^n − 1) / (r − 1).

Step-by-Step Solution:
S_7 = 5 * (3^7 − 1) / (3 − 1) = 5 * (2187 − 1) / 2 = 5 * 2186 / 2 = 5 * 1093 = 5465.

Verification / Alternative check:
List last few terms: day 5 = 405, day 6 = 1215, day 7 = 3645; adding with prior days reaches 5465, matching the formula.

Why Other Options Are Wrong:
₹ 5450, ₹ 5484, ₹ 5486 are near-misses typical of arithmetic slips or truncation; only ₹ 5465 is exact.

Common Pitfalls:
Confusing “amount on the 7th day” (which is 3645) with “total by 7 days” (which is 5465).

Final Answer:
₹ 5465