Question

Find nterm of sireis 1squre+(1squre+2squre)+(1squre+2squre+3squre)+.........

Answer 1

Answer:

$\: \: {1}^{2} + ( {1}^{2} + {2}^{2} ) + ( {1}^{2} + {2}^{2} + {3}^{2} )....n\\ = n. {1}^{2} + (n - 1) {2}^{2} + (n - 3) {3}^{2} ... \\ = \sum(n - k+1) {k}^{2} \\ = \sum((n+1)( {k}^{2} ) - {k}^{3} ) \\ =(n+1) \sum {k}^{2} - \sum {k}^{3} \\ = ( n+1) \times \frac{n(n + 1)(2n + 1)}{6} - ( \frac{n(n + 1)}{2} ) {}^{2} \\ = \frac{ n({n+1})^{2} }{12} (2(2n + 1) - 3(n )) \\ = \frac{ n({n+1})^{2} (n + 2)}{12}$

Answer 2

Step-by-step explanation:

1st term = sum of 1st 1 square

2nd term = sum of 1st 2 squares

nth term = sum of 1st n square

=(n(n+1)/2)^2

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