Let us prove that 12+32+52+⋯+(2n+1)2=3(n+1)(2n+1)(2n+3) using method of mathematical induction.
For n=0 the left side is equal to 11=1, and the right side is equal to 3(0+1)(2⋅0+1)(2⋅0+3)33=1.
Let for n=k the formula 12+32+52+⋯+(2k+1)2=3(k+1)(2k+1)(2k+3) is true.
Let us prove for n=k+1:
12+32+52+⋯+(2k+1)2+(2(k+1)+1)2=3(k+1)(2k+1)(2k+3)+(2k+3)2
=(2k+3)(3(2k2+3k+1+(2k+3))=(2k+3)32k2+3k+1+6k+9=(2k+3)32k2+9k+10
=(2k+3)3(k+2)(2k+5)=3(k+2)(2k+3)(2k+5)=3((k+1)+1)(2(k+1)+1)(2(k+1)+3).
We conclude that this formula is true for any non-negative integer number n.
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