Prove by mathematical induction that 13 33 53 2n 13 n

Prove, by mathematical induction that 1^3 + 3^3 + 5^3 + ... + (2n - 1)^3 = n^2(2n^2 - 1).

Solution

Given that 1³ + 3³ + 5³ +-----------------+ (2n-1)³ = n² (2n²-1) -------- Eq (1)

For n=1:

      1³ = 1² (2.1²-1)

       1=1

For n=k

1³ + 3³ + 5³ +-----------------+ (2k-1)³ = k² (2k²-1)

For n=k+1

1³ + 3³ + 5³ +-----------------+ (2k-1)³ + [2(k+1)-1]³

         = k² (2k²-1) + [2(k+1)-1]³

           = 2k⁴ -k² + [2k+2-1]³

       = 2k⁴ -k² + [2k+1]³

        = 2k⁴ -k² + 8k³ +1 + 12k² + 6k [ (a+b)³ = a³ +b³ + 3a²b+ 3ab²]

      = 2k⁴ + 8k³ + 11k² + 6k +1

(k+1)² [ 2(k+1)²-1]

= (k²+2k+1) [ 2k²+ 4k+ 2-1]

= (k²+2k+1) [ 2k²+ 4k+ 1]

= 2k⁴+ 4k³+ k²+ 4k³+ 8k²+ 2k+ 2k²+ 4k+ 1

= 2k⁴+ 8k³+ 11k² + 6k+ 1

Therefore,

1³ + 3³ + 5³ +-----------------+ (2k-1)³ + [2(k+1)-1]³

= 2k⁴ + 8k³ + 11k² + 6k +1

= (k+1)² [ 2(k+1)²-1]

Hence,

Eq (1) is true for n = k+1

Therefore,

Eq (1) is true for all values of n.