P ( n ) : 1 + 3 + 3 2 + . . . + 3 n − 1 = 3 n − 1 2 The statement P(n) Options: A . &nbsp is true for all natural numbers B . &nbsp is not true for all natural numbers C . &nbsp is true only for a finite number of natural numbers D . &nbsp is true only for all natural numbers greater than 2

Answer: Option A : A P ( n ) : 1 + 3 + 3 2 + . . . + 3 n − 1 = 3 n − 1 2 P ( 1 ) : 1 = 3 − 1 2 ⇒ 1 = 1 − t r u e Assume P(k) is true ⇒ 1 + 3 + 3 2 + . . . + 3 k − 1 = 3 k − 1 2 N o w , P ( k + 1 ) : 1 + 3 + 3 2 + . . . + 3 k − 1 + 3 k = 3 k − 1 2 + 3 k = 3 k ( 1 + 1 2 ) − 1 2 = 3 k + 1 − 1 2 P(k+1) is also true. Hence, P(n) is true for all natural numbers

P(n):1+3+32+...+3n−1=3n−12 The statement P(n)

Question

$P (n) : 1 + 3 + 3^{2} + . . . + 3^{n - 1} = \frac{3^{n} - 1}{2}$
The statement P(n)

Options:

A . is true for all natural numbers

B . is not true for all natural numbers

C . is true only for a finite number of natural numbers

D . is true only for all natural numbers greater than 2

Answer: Option A
:
A

$P (n) : 1 + 3 + 3^{2} + . . . + 3^{n - 1} = \frac{3^{n} - 1}{2} P (1) : 1 = \frac{3 - 1}{2} \Rightarrow 1 = 1 - t r u e$
Assume P(k) is true
$\Rightarrow 1 + 3 + 3^{2} + . . . + 3^{k - 1} = \frac{3^{k} - 1}{2} N o w, P (k + 1) : 1 + 3 + 3^{2} + . . . + 3^{k - 1} + 3^{k} = \frac{3^{k} - 1}{2} + 3^{k} = 3^{k} (1 + \frac{1}{2}) - \frac{1}{2} = \frac{3^{k + 1} - 1}{2}$
P(k+1) is also true.
Hence, P(n) is true for all natural numbers