Correct Answer - Option 3 : 7.5
GIVEN:
Square of sum of four consecutive positive integers is 670 more than the sum of square those four consecutive positive integers.
CALCULATION:
Let consecutive integers are ‘x’, ‘x + 1’, ‘x + 2’ and ‘x + 3’
Square of sum of four consecutive positive integers = [x + (x + 1) + (x + 2) + (x + 3)]2 = (4x + 6)2 = 16x2 + 36 + 48x
Sum of square those four consecutive positive integers = x2 + (x + 1)2 + (x + 22) + (x + 3)2 = x2 + x2 + 1 + 2x + x2 + 4 + 4x + x2 + 9 + 6x = 4x2 + 12x + 14
Now,
(16x2 + 36 + 48x) – (4x2 + 12x + 14) = 670
12x2 + 36x – 648 = 0
x2 + 3x – 54 = 0
x = 6
Required average = [x + (x + 1) + (x + 2) + (x + 3)]/4 = (4x + 6)/4 = x + 1.5 = 6 + 1.5
= 7.5