The value of - -12 22 32 42 22 32 42 52 32 42 52 62 42 52 62 72 equals to

The correct option is A 0 Δ =1^2 amp; 2^2 amp; 3^2 amp; 4^2 2^2 amp; 3^2 amp; 4^2 amp; 5^2 3^2 amp; 4^2 amp; 5^2 amp; 6^2 ...

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Properties of Determinants

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The value of $Δ = ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} 1^{2} & 2^{2} & 3^{2} & 4^{2} 2^{2} & 3^{2} & 4^{2} & 5^{2} 3^{2} & 4^{2} & 5^{2} & 6^{2} 4^{2} & 5^{2} & 6^{2} & 7^{2} \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣$ equals to

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1^{2} + 2^{2} + 3^{2} + 4^{2} + . . . + 7^{2}

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Δ = ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} 1^{2} & 2^{2} & 3^{2} & 4^{2} 2^{2} & 3^{2} & 4^{2} & 5^{2} 3^{2} & 4^{2} & 5^{2} & 6^{2} 4^{2} & 5^{2} & 6^{2} & 7^{2} \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣

S = \frac{1}{1^{2}} + \frac{1}{2^{2}} + \frac{1}{3^{2}} + \frac{1}{4^{2}} + . . . . . . . . . . . . .

\infty

\frac{1}{1^{2}} + \frac{1}{3^{2}} + \frac{1}{5^{2}} + \frac{1}{7^{2}} + . . . . . . . . . . . . .

\infty

1^{2} + 2^{2} + 3^{2} + 4^{2} + . . . + 7^{2} =

1^{2} = \frac{1}{6} [1 \times (1 + 1) \times (2 \times 1 + 1)] 1^{2} + 2^{2} = \frac{1}{6} [2 \times (2 + 1) \times (2 \times 2 + 1)] 1^{2} + 2^{2} + 3^{2} = \frac{1}{6} [3 \times (3 + 1) \times (2 \times 3 + 1)] 1^{2} + 2^{2} + 3^{2} + 4^{2} = \frac{1}{6} [4 \times (4 + 1) \times (2 \times 4 + 1)]

\frac{1}{1^{2}} + \frac{1}{2^{2}} + \frac{1}{3^{2}} + \frac{1}{4^{2}} + . . . . \infty = \frac{π^{2}}{6}

\frac{1}{1^{2}} + \frac{1}{3^{2}} + \frac{1}{5^{2}} + . . . . \infty

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