Question

1/5²=1×2²/5²×2²=_____=_______
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Answer 1

Sη=1²+2²+3²+…..+n²…………..(1)

we know that ,

(a-1)³=a³-3a²+3a-1

=>3a²-3a+1=a³-(a-1)³………….(2)

now putting a=1,2,3,….,n we get,

3.1²-3.1+1=1³-0³

3.2²-3.2+1=2³-1³

3.3²-3.3+1=3³-2³

…………………..

…………………..

3(n-1)²-3(n-1)+1=(n-1)³-(n-2)³

3n²-3n+1=n³-(n-1)³

adding all the n equation ,we get,

3(1²+2²+3²+….+n²)-3(1+2+3+….+n)+n=n³

=>3Sη-3{n(n+1)}/2+n=n³

=>3Sη=n³+[3{n(n+1)}/2]-n

=>3Sη=n³+{3n²+3n-2n}/2

=>3Sη={2n³+3n³+n}/2

=>3Sη=n{2n²+3n+1}/2

=>3Sη=n{2n²+2n+n+1}/2

=>3Sη=n{2n(n+1)+1(n+1)}/2

=>3Sη=n(n+1)(2n+1)/2

=>Sη=n(n+1)(2n+1)/6