Question:medium

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

Updated On: Jan 15, 2026
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Correct Answer: 548

Solution and Explanation

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

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

The objective is to determine the average of all 3-digit numbers within this AP. The smallest 3-digit number in the AP, which is ≥ 100, is: \[ a = 106 \]. The largest 3-digit number in the AP, which is ≤ 999, is: \[ l = 990 \]

Step 1: Determine the number of terms (n)

Employing the formula: \[ n = \frac{l - a}{d} + 1 = \frac{990 - 106}{17} + 1 = \frac{884}{17} + 1 = 52 + 1 = 53 \]. However, as 884 is exactly divisible by 17, resulting in 52, the accurate count of terms is: \[ n = 52 \]

Step 2: Calculate the sum using the sum formula

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

Step 3: Compute the average

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

Final Answer:

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

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