Please assist me with the math problem

Question

Use the mathematical induction to prove that the following statement is true for every positive integer n

1+8+27+...(n² (n+1)^2)/4

Answer 1

Note that your formula as submitted is incorrect: the 4 belongs in the numerator of the expression, not in the exponent.

The statement is trivially true for n=1

Assume it is true for n

Add (n+1)³ to the formula for n to get

[(n⁴ + 2n³ + n²) + 4(n³ + 3n² + 3n +1)]/4

Expand and collect the terms to get (n⁴ + 6n³ + 13n² + 12n + 4)/4

Divide the numerator by n²+ 2n + 1 to get n² + 4n + 4

Therefore the sum to the term in n+1 is (n+1)² (n+2)²/4 which is the formula with n+1 substituted for n.

Q.E.D. by induction!