P n - 1-3-32-3 n -1-3 n -1-2The statement Pn A- is true only for a finite number of natural numbersB- is true only for all natural numbers greater than 2C- is true for all natural numbersD- is not true for all natural numbers

The correct option is C is true for all natural numbers P(n):1+3+3^2+...+3^n 1=3^n 1/2 P(1): 1=3 1/2⇒ 1=1 true Assume P(k) is true ⇒ 1+3+3^2+...+3^k 1=3^k ...

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Byju's Answer

Standard XI

Mathematics

Proof by mathematical induction

P n : 1+3+32+...

Question

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

is true for all natural numbers

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is not true for all natural numbers

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is true only for a finite number of natural numbers

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is true only for all natural numbers greater than 2

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Solution

The correct option is A
is true for all natural numbers

$P (n) : 1 + 3 + 3^{2} + . . . + 3^{n - 1} = \frac{3^{n} - 1}{2} P (1) : 1 = \frac{3 - 1}{2} \Rightarrow 1 = 1 - t r u e$
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

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

The correct option is A is true for all natural numbers P(n):1+3+32+...+3n−1=3n−12P(1):1=3−12⇒1=1− true Assume P(k) is true ⇒1+3+32+...+3k−1=3k−12Now, P(k+1):1+3+32+...+3k−1+3k=3k−12+3k=3k(1+12)−12=3k+1−12 P(k+1) is also true. Hence, P(n) is true for all natural numbers

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