Show 7 2 n - 1 is divisible by 48 for all n ∈ N .

Let P n : 7 2 n - 1 is divisible by 48 . Now, P 1 ≡ 7 2 - 1 = 48 Hence, P 1 is true. Let P k is true, then 7 2 k - 1 = 48 λ , λ ∈ N Now, P k + 1 : 7 2 k + 2 - 1 ⇒ P k + 1 : 49 · 7 2 k - 49 + 48 ⇒ P k + 1 : 49 7 2 k - 1 + 48 ⇒ P k + 1 : 49 × 48 λ + 48 ⇒ P k + 1 : 48 49 λ + 1 Hence, P k + 1 is also true when P k is true. So, P n is true for all n ∈ N .

E M B I B E

Gamma Question Bank for Engineering Mathematics>Chapter 10 - Principle of Mathematical Induction>Assignment>Q 26

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IMPORTANT

Earn 100

Show that $4^{n} + 15 n - 1$ is divisible by $9 .$

Important Questions on Principle of Mathematical Induction

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IMPORTANT

Gamma Question Bank for Engineering Mathematics>Chapter 10 - Principle of Mathematical Induction>Assignment>Q 27

Show

7^{2 n} - 1

is divisible by

48

for all

n \in N

.

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IMPORTANT

Gamma Question Bank for Engineering Mathematics>Chapter 10 - Principle of Mathematical Induction>Assignment>Q 28

Using

P M I

, show that

5^{2 n + 2} - 24 n - 25

is divisible by

576

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Gamma Question Bank for Engineering Mathematics>Chapter 10 - Principle of Mathematical Induction>Assignment>Q 29

Using

P M I

, prove

6^{n + 2} + 7^{2 n + 1}

is divisible by

43 .

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Gamma Question Bank for Engineering Mathematics>Chapter 10 - Principle of Mathematical Induction>Assignment>Q 30

Using

P M I

, prove

12^{n} + 25^{n - 1}

is divisible by

13

.

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Gamma Question Bank for Engineering Mathematics>Chapter 10 - Principle of Mathematical Induction>Assignment>Q 31

Prove by the method of induction that every even power of every odd integer greater than

1,

When divided by

8

leaves the remainder

1 .

EASY

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IMPORTANT

Gamma Question Bank for Engineering Mathematics>Chapter 10 - Principle of Mathematical Induction>Assignment>Q 1

Let

P (n)

be a statement and let

P (k) \Rightarrow P (k + 1)

, for some natural number

k

, then

P (n)

is true for all

n \in N

.

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IMPORTANT

Gamma Question Bank for Engineering Mathematics>Chapter 10 - Principle of Mathematical Induction>Assignment>Q 2

The statement $(n + 3)^{2} > 2^{n + 3}$ is true.

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IMPORTANT

Gamma Question Bank for Engineering Mathematics>Chapter 10 - Principle of Mathematical Induction>Assignment>Q 3

A student was asked to prove a statement $P(n)$ by using the principal of mathematical induction. He proved that $P (n) \Rightarrow P (n + 1)$ for all $n \in N$ and also that $P (4)$ is true: