Question

Use mathematical induction to prove the statement is true for all positive integers n.

10 + 30 + 60 + ... + 10n = 5n(n + 1)

Answer 1

Call the statement

10 + 30 + 60 + ... + 10n = 5n(n + 1)

P(n), and show using induction the statement is true for all positive integers n.

Basis: show the statement holds true for n=0

10 (0) = 5*0(0+1)

0 = 0

The basis holds for P(0)

Inductive Hypothesis:

If P(k) holds for some unspecified value of k, then so does P(k+1); it must be shown that

(10 + 30 + 60 ..... + 10k) + 10(k+1) = 5(k+1)((k+1)+1)

Using the inductive hypothesis that P(k+1) holds, we can say that the left hand can be written as

5k(k+1) + 10(k+1) = 5k^2 + 5k + 10k +10

= 5k^2 + 10k + 5 + 5k + 5

= 5(k^2 + 2k +1 + k + 1)

= 5((k^2+2k +1)) +(k+1))

= 5((k + 1)(k + 1) + (k + 1))

= 5(k+1)((k+1) + 1)

Therefor P(k+1) holds algebraically, and since the basis and inductive step have been performed, the statement P(n) holds for all positive integers.