About prove each of the following statements using mathematical induction. (a) define the sequence {cn} as follows: c0 = 5 ck = (ck-1)2 for k ≥ 1 prove that for n ≥ 0, cn = 52n. note that in the explicit formula for cn, the exponent of 5 is 2n. (b) define the sequence {bn} as follows: b0 = 1 bk = 2bk-1 + 1 for k ≥ 1 prove that for n ≥ 0, bn = 2n+1 - 1.

a.

For , we have by this definition which is equal to for .

Assume for some . We want to use this to show that this implies . By the recursive definition,

and we're done.

b.

For , we have , and .

Assume for some . Then for , we have

and we're done.

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5/12, or 5: 12

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