Use mathematical induction to prove that the statement is true for every positive integer n.10 + 20 + 30 + . . . + 10n = 5n(n + 1)
What will be an ideal response?
| S1: | 10  (5 ? 1)(1 + 1) |
5 ? 210 = 10 ?
Sk: 10 + 20 + 30 + . . . + 10k = 5k(k + 1)
Sk+1: 10 + 20 + 30 + . . . + 10(k + 1) = 5(k + 1)(k + 2)
We work with Sk. Because we assume that Sk is true, we add the next multiple of 10, namely
10(k+1), to both sides.
10 + 20 + 30 + . . . + 10k + 10(k + 1) = 5k(k + 1) + 10(k + 1)
10 + 20 + 30 + . . . + 10(k + 1) = (k + 1)(5k + 10)
10 + 20 + 30 + . . . + 10(k + 1) = 5(k + 1)(k + 2)
We have shown that if we assume that Sk is true, and we add (10(k+1) to both sides of Sk, then Sk+1 is also true. By the principle of mathematical induction, the statement Sn is true for every positive integer n.
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