Question

The average of all 3-digit terms in the arithmetic progression 38,55,72,...,is

Answer 1

Correct Answer: 548

Given arithmetic progression (AP): 38, 55, 72, ...

The common difference is: \[ d = 55 - 38 = 17 \]

We want to find the average of all 3-digit numbers in this AP. The smallest 3-digit number ≥ 100 that fits the AP is: \[ a = 106 \] The largest 3-digit number ≤ 999 in the AP is: \[ l = 990 \]

Step 1: Find the number of terms (n)

Use the formula: \[ n = \frac{l - a}{d} + 1 = \frac{990 - 106}{17} + 1 = \frac{884}{17} + 1 = 52 + 1 = 53 \] But since 884 ÷ 17 = 52, the correct value is: \[ n = 52 \]

Step 2: Use the sum formula

The sum of the AP terms is: \[ S_n = \frac{n}{2} \cdot (a + l) = \frac{52}{2} \cdot (106 + 990) = 26 \cdot 1096 = 28496 \]

Step 3: Find the average

\[ \text{Average} = \frac{S_n}{n} = \frac{28496}{52} = 548 \]

Final Answer:

\[ \boxed{548} \] The average of all 3-digit terms in the arithmetic progression is 548.