Does anyone feel like solving some mathematical proofs by induction?

by

A) 2 + 4 + 6+ … + 2n = n (n + 1)

B) 1 + 2^1 + 2^2 + … + 2^n = 2^(n+1)-1

C) 1^2 + 2^2 + 3^2 + … + n^2 = (n (n + 1)(2n + 1)) / 6

Let P(n): 1^3 + 2^3 + 3^3 + … + n^3 = (n^2 (n + 1)^2 + 4) / 4
Da) use P(k) to show P(k + 1)
Db) Is P(n) true for all n >= 1

Thank you in advance!

✅ Answers

? Best Answer

  • A) check for n=1: 2=1(1+1)=2

    We assume the formula is correct for ‘n’: 2 + 4 + 6+ … + 2n = n (n + 1)
    Now, the induction step, prove it holds for (n+1)
    2+4+…+2n + 2(n+1) = (n+1)(n+2)
    n(n+1) + 2(n+1) = (n+1)(n+2) –> divide by (n+1)
    n + 2 = n + 2 –> Correct.

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