Question

Prove the following by using the principle of mathematical induction where n N :

2 3 1 3 1 1 3 3 3 .......... 3 .

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Answer 1

Let the given statement be P(n) , i.e.,

P(n): 1 + 3 + 32+ ………..+3n-1 = (3n – 1)/2

For n = 1, we have

P(1): (31 – 1)/2 = 3-1/2 = 2/2 = 1 , which is true.

Let P(k) be true for some positive integer k, i.e.,

1 + 3 + 32+ ………..+3k-1 = (3k – 1)/2

We shall prove that P(k+1) is true.

Consider

1 + 3 + 32+ ………..+3n-1 + 3(k+1)-1 = 1 + 3 + 32+ ………..+3k-1 + 3k

=(3k – 1)/2 +3k

=[(3k – 1)+2.3k]/2

=[(1+2)3k – 1]/2

= [3.3k – 1]/2

= [3k+1 – 1]/2

Thus, P(k + 1) is true whenever P(k) is true.

Answer 2

Step-by-step explanation:

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