CMAT 2016 Solved Question Paper

Question : 7 of 100

Marks: +1, -0

Teacher wrote N natural numbers, starting with 1 to N on the blackboard. Ashish finds the sum of squares of all those numbers written on the board and Aayushi calculated the square of the sum of all the numbers written on the board. The ratio of the answers obtained by Ashish and Aayushi is 17 : 325. Find N.

Validate

Solution:

(a) Sum calculated by Ashish =

1^2 + 2^ 2 + .......... + N^2

{N [N + 1) (2N + 1)}/6

Sum calculated by Aayushi =

(1+2+...........+N)^2

[{N(N + 1)}/2]^2

[{N^2(N + 1)^2}/4]

∴

\;\;\;\;{{N(N + 1)(2N+1)}/6}/{{N^2(N + 1)^2}/4}

{17}/{325}

⇒ 650 (2N + 1) = 51 (

N^2

+ N)
⇒ 1300 N + 650= 51

N^2

+ 51N
⇒ 51

N^2

-1249 N - 650 = 0
⇒ 51

N^2

-1275N + 26N -650 = 0
⇒ 51N (N-25) +26 (N-25) = 0
⇒ (51N +26) (N - 25) = 0
⇒ N = 25

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