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If Sk=(1+2+3+...k)/k then find the value...

If `S_k=(1+2+3+...k)/k` then find the value of `S_1^2+S_2^2+....S_n^2`

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The correct Answer is:
D
Since, `S_(k) = (1 + 2 + 3+ ..+ k)/(k)`
`= (k (k +1))/(2k) = (k +1)/(2)`
So, `S_(k)^(2) = ((k+1)/(2))^(2) = (1)/(4) (k+1)^(2)` ...(i)
Now, `(5)/(12) A = S_(1)^(2) + S_(2)^(2) + S_(3)^(2) + ....S_(10)^(2) = underset(k=1)overset(10)sum S_(k)^(2)`
`rArr (5)/(12) A = (1)/(4) underset(k =1)overset(10)sum (k +1)^(2) = (1)/(4) [2^(2) + 3^(2) + 4^(2) + ...11^(2)]`
`= (1)/(4) [(11 xx(11 + 1) (2xx 11 + 1))/(6) - 1^(2)]`
`[ :' sum n^(2) = (n (n+1) (2n+1))/(6)]`
`= (1)/(4) [(11xx 12 xx 23)/(6) -1] = (1)/(4) [(22 xx 23) -1]`
`= (1)/(4) [506 -1] = (1)/(4) [505]`
`rArr (5)/(12) A = (505)/(4) rArr A = 303`
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