Answer the following: Prove by method of induction 52n − 22n is divisible by 3, for all n ∈ N

प्रश्न

Answer the following:

Prove by method of induction 5²ⁿ − 2²ⁿ is divisible by 3, for all n ∈ N

योग

उत्तर

Let P(n) ≡ 5²ⁿ – 2²ⁿ is divisible by 3, for all n ∈ N.

Step 1:

For n = 1, 5²ⁿ – 2²ⁿ = 5² – 2² = 25 – 4 = 21, which is divisible by 3.

∴ P(1) is true.

Step 2:

Let us assume that for some k ∈ N, P(k) is true, i.e. 5^2k – 2^2k is divisible by 3.

∴ `(5^(2"k") - 2^(2"k"))/3` = m (Say), whre m ∈ N

∴ 5^2k – 2^2k = 3m

∴ 5^2k = 2^2k + 3m ...(1)

Step 3:

To prove that P(k + 1) is true, i.e., to prove that `5^(2("k" + 1)) - 2^(2("k" + 1))` is divisible by 3.

Now, `5^(2("k" + 1)) - 2^(2("k" + 1))` = 5^2k+2 – 2^2k+2

= 5^2k .5² – 2^2k . 2²

= (2^2k + 3m)25 – 2^2k . 4 ...[By (1)]

= 25(2^2k) + 75m – 4(2^2k)

= 21(2^2k) + 75m

= 3[7.2^2k + 25m]

∴ `(5^(2("k" + 1)) - 2^(2("k"+1)))/3` = 7.2^2k + 25m, where (7.2^2k + 25m) ∈ N

∴ `5^(2("k" + 1)) - 2^(2("k" + 1))` is divisible by 3

∴ P(k + 1) is true.

Step 4:

From all the above steps and by the principle of mathematical induction P(n) is true for all n ∈ N,

i.e., 5²ⁿ – 2²ⁿ is divisible by 3, for all n ∈ N.

shaalaa.com

Principle of Mathematical Induction

क्या इस प्रश्न या उत्तर में कोई त्रुटि है?

Q II. (11) (iii)Q II. (11) (ii)Q II. (12)

अध्याय 4: Methods of Induction and Binomial Theorem - Miscellaneous Exercise 4.2 [पृष्ठ ८६]

APPEARS IN

बालभारती Mathematics and Statistics (Arts and Science) Part 2 [English] Standard 11 Maharashtra State Board

अध्याय 4 Methods of Induction and Binomial Theorem
Miscellaneous Exercise 4.2 | Q II. (11) (iii) | पृष्ठ ८६

संबंधित प्रश्न

Prove the following by using the principle of mathematical induction for all n ∈ N:

1.3 + 3.5 + 5.7 + ...+(2n -1)(2n + 1) = `(n(4n^2 + 6n -1))/3`

Prove the following by using the principle of mathematical induction for all n ∈ N:

`1^2 + 3^2 + 5^2 + ... + (2n -1)^2 = (n(2n - 1) (2n + 1))/3`

Prove the following by using the principle of mathematical induction for all n ∈ N: 3²ⁿ^{+ 2} – 8n– 9 is divisible by 8.

Prove the following by using the principle of mathematical induction for all n ∈ N: 41ⁿ – 14ⁿ is a multiple of 27.

1² + 2² + 3² + ... + n² =\[\frac{n(n + 1)(2n + 1)}{6}\] .

\[\frac{1}{1 . 4} + \frac{1}{4 . 7} + \frac{1}{7 . 10} + . . . + \frac{1}{(3n - 2)(3n + 1)} = \frac{n}{3n + 1}\]

1.3 + 2.4 + 3.5 + ... + n. (n + 2) = \[\frac{1}{6}n(n + 1)(2n + 7)\]

1.3 + 3.5 + 5.7 + ... + (2n − 1) (2n + 1) =\[\frac{n(4 n^2 + 6n - 1)}{3}\]

3²ⁿ⁺² −8n − 9 is divisible by 8 for all n ∈ N.

11ⁿ⁺² + 12²ⁿ⁺¹ is divisible by 133 for all n ∈ N.

Prove that n³ - 7n + 3 is divisible by 3 for all n \[\in\] N .

Prove that 1 + 2 + 2² + ... + 2ⁿ = 2ⁿ⁺¹ - 1 for all n \[\in\] N .

Let P(n) be the statement : 2ⁿ ≥ 3n. If P(r) is true, show that P(r + 1) is true. Do you conclude that P(n) is true for all n ∈ N?

\[\text{ Prove that } \frac{1}{n + 1} + \frac{1}{n + 2} + . . . + \frac{1}{2n} > \frac{13}{24}, \text{ for all natural numbers } n > 1 .\]

\[\text{ Let } P\left( n \right) \text{ be the statement } : 2^n \geq 3n . \text{ If } P\left( r \right) \text{ is true, then show that } P\left( r + 1 \right) \text{ is true . Do you conclude that } P\left( n \right)\text{ is true for all n } \in N?\]

\[\text { A sequence } x_1 , x_2 , x_3 , . . . \text{ is defined by letting } x_1 = 2 \text{ and } x_k = \frac{x_{k - 1}}{k} \text{ for all natural numbers } k, k \geq 2 . \text{ Show that } x_n = \frac{2}{n!} \text{ for all } n \in N .\]

\[\text{ A sequence } x_0 , x_1 , x_2 , x_3 , . . . \text{ is defined by letting } x_0 = 5 and x_k = 4 + x_{k - 1}\text{ for all natural number k . } \]
\[\text{ Show that } x_n = 5 + 4n \text{ for all n } \in N \text{ using mathematical induction .} \]