Use mathematical induction to prove that the statement is true for every positive integer n.1 ? 4 + 2 ? 4 + 3 ? 4 + . . . + 4n = 
What will be an ideal response?
| S1: | 1 ? 4   |
4 = 4 ?
Sk: 1 ? 4 + 2 ? 4 + 3 ? 4 + . . . + 4k =
Sk+1: 1 ? 4 + 2 ? 4 + 3 ? 4 + . . . + 4(k + 1) =
We work with Sk. Because we assume that Sk is true, we add the next consecutive term, namely
4(k+1), to both sides."
1 ? 4 + 2 ? 4 + 3 ? 4 + . . . + 4k + 4(k + 1) =
+ 4(k + 1)1 ? 4 + 2 ? 4 + 3 ? 4 + . . . + 4(k + 1) =
+ 
1 ? 4 + 2 ? 4 + 3 ? 4 + . . . + 4(k + 1) =
1 ? 4 + 2 ? 4 + 3 ? 4 + . . . + 4(k + 1) =
We have shown that if we assume that Sk is true, and we add (4(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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