DO NOT USE A CALCULATOR TO OBTAIN YOUR ANSWERS W CABLE LEAVE YOUR ANSWERS IN TER

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DO NOT USE A CALCULATOR TO OBTAIN YOUR ANSWERS W CABLE LEAVE YOUR ANSWERS IN TERMS OF FACTORIAIS. C Hence one 1. Suppose ompute (a) ) A +B (n) BC 2. Use mathematical imduction to show th al (a) 1+34, + (2n + 1) = (n+1for all n E N. n 21. (b) 2 for all nEN n21 3. Consider numbers (strings) of length 5 over the tee digits 0,123 repetition of digits is allowed. The mnbers niay begin with (a) How many different numbers are there (b) How many of these numbers have no (e) How many of these numbes have at least one

Explanation / Answer

To prove that n < 2n for all positive integers n.

n < 2n for all n N ; n 1

Solution:

Let P(n) : n < 2n

For n =1

         1 < 21

i.e. 1 < 2

hence P(1) is true.

Inductive Step :

Assume P(k) holds, i.e. , k < 2k , for any positive integer k.

P(k+1) will also hold when P(k) is true.

As, k < 2k

2k < 2 . 2k (Multiplying both sides by 2)

2k < 2k+1

k+k < 2k+1

But we know for k>1, k+1 < k+k

Therefore, k+1 < 2k+1

Hence this is also true for P(k+1)

By mathematical induction n<2n for all n N ; n 1

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