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

Question

UPSC Quizroom · Accepted Answer

Step 1: Identify the arithmetic progression (AP)

Given AP: 38, 55, 72, …

First term, a1 = 38

Common difference, d = 55 - 38 = 17

Step 2: Find the first 3-digit term

Let the n-th term be the first 3-digit term:
a_n = a1 + (n - 1) * d ≥ 100

Substitute values:
38 + (n - 1) * 17 ≥ 100
17 * (n - 1) ≥ 62
n - 1 ≥ 62 / 17 ≈ 3.647
n ≥ 4.647 → n = 5

Check: 5th term = 38 + 4*17 = 38 + 68 = 106 ✅ 3-digit number

So first 3-digit term = 106

Step 3: Find the last 3-digit term

Let the last 3-digit term be a_m ≤ 999
a_m = a1 + (m - 1) * d ≤ 999

Substitute values:
38 + (m - 1) * 17 ≤ 999
17 * (m - 1) ≤ 961
m - 1 ≤ 961 / 17 ≈ 56.529
m = 57

Check: 57th term = 38 + 56*17 = 38 + 952 = 990 ✅ 3-digit number

58th term = 38 + 57*17 = 1007 ❌ not 3-digit

So last 3-digit term = 990
Step 4: Average of all 3-digit terms

Average of all terms in an AP = (first term + last term) / 2

Average = (106 + 990) / 2 = 1096 / 2 = 548

Answer: 548

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