Shortcut Trick
Average of first n natural numbers = (n + 1) ÷ 2
Average of first n perfect squares = (n + 1)(2n + 1) ÷ 6
For n = 5: A = (5 + 1) ÷ 2 = 3
For n = 5: B = (5 + 1)(2 × 5 + 1) ÷ 6 = (6 × 11) ÷ 6 = 11
Average of A and B = (3 + 11) ÷ 2 = 7
∴ The correct answer is 7.
Alternate Method
Given:
First 5 natural numbers: 1, 2, 3, 4, 5
First 5 perfect squares: 12, 22, 32, 42, 52 → 1, 4, 9, 16, 25
Calculations:
⇒ A = (1 + 2 + 3 + 4 + 5) ÷ 5
⇒ A = 15 ÷ 5 = 3
⇒ B = (1 + 4 + 9 + 16 + 25) ÷ 5
⇒ B = 55 ÷ 5 = 11
⇒ Average of A and B = (A + B) ÷ 2
⇒ Average = (3 + 11) ÷ 2 = 14 ÷ 2 = 7
∴ The correct answer is 7.
Additional Information
Sum of first n natural numbers
The sum is given by the formula S = n(n + 1) ÷ 2.
Sum of squares of first n natural numbers
The sum is given by the formula S = n(n + 1)(2n + 1) ÷ 6.
Average of Arithmetic Progression
For any AP, the average is (First term + Last term) ÷ 2.