The mean of n observations

1, 4, 9, 16 ...., n2 is 130. What is the value of n?

This question was previously asked in
NDA-II 2024 (Maths) Official Paper (Held On: 01 Sept, 2024)
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  1. 18
  2. 19
  3. 20
  4. 21

Answer (Detailed Solution Below)

Option 2 : 19
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Explanation:

Mean  = \(\frac{1+ 4+ 9 ... n^2}{n}\)

⇒ 130 = \(\frac{(n(n+1)(2n+1)}{6n}\)

⇒ 780 = 2n+ 3n + 1

⇒ 2n2 + 3n – 779 =0

⇒ 2n2 + 41n – 38n – 779 = 0

⇒ n(2n + 41) – 19(2n + 41) = 0 

⇒ (2n + 41) (n – 19) = 0

⇒ x = 19 ..... ( n = \(-\frac{41}{2}\) is not possible)

∴ Option (b) is correct.

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