Question

Each one of 100 boxes is filled with 50 one-rupee coins on the first day and 25 more coins are added every next day. The Arithmetic Progression (AP) representing this situation is

• A. \( 100, 50, 25, 10, \dots \)
• B. \( 50, 25, 25, 25, \dots \)
• C. \( 50, 75, 100, 125, \dots \)
• D. \( 50, 25, 75, 100, \dots \)

Hint:

An arithmetic progression is a sequence of numbers such that the difference between consecutive terms is constant. This constant difference is called the common difference.

Answer 1

The Correct Option is C

Step 1: Identify the first term.
On the first day, each box is filled with 50 coins. So, the first term of the arithmetic progression is \( a_1 = 50 \). Step 2: Identify the common difference.
Every next day, 25 more coins are added to each box. This means the common difference \( d \) of the arithmetic progression is \( d = 25 \). Step 3: Write the terms of the arithmetic progression.
The terms of an arithmetic progression are given by \( a_n = a_1 + (n-1)d \). The first few terms are:
Day 1: \( a_1 = 50 \)
Day 2: \( a_2 = a_1 + d = 50 + 25 = 75 \)
Day 3: \( a_3 = a_1 + 2d = 50 + 2(25) = 50 + 50 = 100 \)
Day 4: \( a_4 = a_1 + 3d = 50 + 3(25) = 50 + 75 = 125 \)
The arithmetic progression representing the number of coins in each box each day is \( 50, 75, 100, 125, \dots \) Step 4: Match the AP with the given options.
Comparing our AP with the options, we find that option (3) matches: \( 50, 75, 100, 125, \dots \)