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Prove that 1^2 + (1^2 + 2^2) + (1^2 + 2^2 + 3^2) + ... n terms = n/12 (n + 1)^2 (n + 2)

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Your problem `->` Prove that 1^2 + (1^2 + 2^2) + (1^2 + 2^2 + 3^2) + ... n terms = n/12 (n + 1)^2 (n + 2)


L.H.S. `= 1 + (1 + 2) + (1 + 2 + 3) + ... + n` terms

`= sum [ f(n) ]`

`= sum [ sum n ]`

`= sum [ n/2 (n + 1) ]`

`= 1/2 sum n^2 + 1/2 sum n`

`= 1/2 * (n (n + 1) (2n + 1))/6 + 1/2 * (n (n + 1))/2`

`= (n (n + 1) (2n + 1))/12 + (n (n + 1))/4`

`= (n (n + 1) (2n + 1 + 3))/12`

`= (n (n + 1) (2n + 4))/12`

`= (n (n + 1) (n + 2))/6`

`=` R.H.S. (Proved)






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