Discussion

Explanation:

38, 55, 72, … forms an AP whose first term is 38 and common difference is 17.
∴ T_n = 38 + (n - 1) × 17

To find the average we need to find the highest and lowest 3-digit numbers of this sequence.

Lowest: 38 + (n - 1) × 17 > 99
⇒ 17n - 17 > 61
⇒ 17n > 78
⇒ n > 4.58
∴ Least possible value of n = 5
⇒ Least such number = 38 + 4 × 17 = 106

Highest: 38 + (n - 1) × 17 < 999
⇒ 17n - 17 < 961
⇒ 17n < 978
⇒ n < 56.5
∴ Highest possible value of n = 56
⇒ Highest such number = 38 + 56 × 17 = 990

∴ The average of the sequence (AP) is same as the average of lowest and highest terms = $\frac{106+990}{2}$ = 548

Hence, 548.

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