M2 Mathematical Induction

3a) When n=1,
RHS = 1^3 = 1
LHS = 1^2 * (2*1^2 - 1) = 1 * 1 = 1
therefore, P(1) is true
Assume P(k) is true, ie
1^3 + 2^3 + . . . + (2k - 1)^3 = k^2 * (2k^2 - 1)
When n = k + 1
1^3 + 2^3 + . . . + (2k - 1)^3 + (2k + 1)^3
= k^2 * (2k^2 - 1) + (2k + 1)^3
= 2k^4 - k^2 + 8k^3 + 12k^2 + 6k + 1
= 2k^4 + 8k^3 + 11l^2 + 6k + 1
= (k + 1)(2k^3 + 6k^2 + 5k + 1)
= (k + 1)(k + 1)(2k^2 + 4k + 1)
= (k + 1)(k + 1)[2(k^2 + 2k + 1) - 1]
= (k + 1)^2 * [2(k + 1)^2 - 1]
therefore, P(k + 1) is also true.

Conclusion : by Principle of MI, P(n) is true for all positive integers n.

3b)
2^3 + 6^3 + 10^3 + . . . + 38^3
= 2^3 * (1^3 + 3^3 + 5^3 + . . . + 19^3)
= 2^3 * [10^2 * (2*10^2 -1)] . . . . . [from (a), put n = 10]
= 8 * 100 * 199
= 159200