First we need to show the result is true for smallest value of n, in this case n = 0/
When n =0 the left side has only one term n2 = 0
The right side is
6n(n+2)(2n+1)=60(0+1)(0+1)=0
The second step is to show the statement is true for n = k, k+1 and so on.
Suppose, that statement is true for n = K.
12 + 22 + 32 + … + k2 = 6k(k+1)(2k+1)
We need to prove, that statement is true fof n = k+1
12 + 22 + 32 + … + (k+1)2 = 6(k+1)(k+2)(2(k+1)+1)
12 + 22 + 32 + … +k2 + (k+1)2 = 6(k+1)(k+2)(2(k+1)+1)
We know, that
12 + 22 + 32 + … +k2 = 6k(k+1)(2k+1)
So,
6k(k+1)(2k+1)+(k+1)2=6k(k+1)(2k+1)+6(k+1)2
=6k+1(k(2k+1)+6(k+1))
=6k+1(2k2+k+6k+6)
=6k+1(2k2+7k+6)
2k2+7k+6=0
2k2+4k+3k+6=0
2k(k+2)+3(k+2)=0
(2k+3)(k+2)=0
frack+16(2k2+7k+6)=6(k+1)(k+2)(2k+3)
=6(k+1)((k+1)+1)(2(k+1)+1) = RHS hence proved
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