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Square of sum of four consecutive positive integers is 670 more than the sum of square those four consecutive positive integers, then find the average of the four numbers.
1. 9
2. 12.5
3. 7.5
4. 6

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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

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