Prove 7^{k + 1} is divisible by 6.

Assume 7^k+1 – 1 is divisible by 6.

Assume 7¹–1 is divisible by 6.

Assume 7^k – 1 is divisible by 6.

The base case is true.

The statement is true for all n.

The statement is false for some n.

The statement is true for n = k.

Assume P(k) = 3^{2k + 2} – 8k – 9 is divisible by 8, and prove for k .

Assume P(k) = 3^{2k + 2} – 8k – 9 is divisible by 7, and prove for k + 1.

Assume P(k) = 3^{2k + 2} – 8k – 9 is divisible by 6, and prove for k + 1.

Assume P(k) = 3^{2k + 2} – 8k – 9 is divisible by 8, and prove for k + 1.

Show 1 + 3 + 5 + . . . + (2k – 1) = k(k + 1) .

Show 1 + 3 + 5 + . . . + (2k + 1) = (k + 1)².

Show 1 + 3 + 5 + . . . + (2k + 1) = 2k².

Show 1 + 3 + 5 + . . . + (2k – 1) = k².

Prove for

Prove for

Prove for

Prove for

–i

–1

0

1

9

13

16

25