3^(n - 1) %2B ldots %2B 1 %2B 3 %2B 3^2=(3^n - 1)/2

1.	3^(n - 1) %2B ldots %2B 1 %2B 3 %2B 3^2=(3^n - 1)/2
Answer» 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. Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., N.

Answer»

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.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., N.

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