Advertisements
Advertisements
प्रश्न
72n + 23n−3. 3n−1 is divisible by 25 for all n ∈ N.
उत्तर
Let P(n) be the given statement.
Now,
\[P(n): 7^{2n} + 2^{3n - 3} . 3^{n - 1} \text{ is divisible by } 25 . \]
\[\text{ Step1} : \]
\[P(1): 7^2 + 2^{3 - 3} . 3^{1 - 1} = 49 + 1 = 50 \]
\[\text{ It is divisible by 25} . \]
\[\text{ Thus, P(1) is true} \]
\[\text{ Step2: Let } P\left( m \right) \text{ be true . } \]
\[\text{ Now } , \]
\[ 7^{2m} + 2^{3m - 3} . 3^{m - 1} \text{ is divisible by} 25 . \]
\[Suppose: \]
\[ 7^{2m} + 2^{3m - 3} . 3^{m - 1} = 25\lambda . . . (1)\]
\[\text{ We have to show that } P\left( m + 1 \right) \text{ is true whenever P(m) is true} . \]
\[\text{ Now, } \]
\[P\left( m + 1 \right) = 7^{2m + 2} + 2^{3m} . 3^m \]
\[ = 7^{2m + 2} + 7^2 . 2^{3m - 3} . 3^{m - 1} - 7^2 . 2^{3m - 3} . 3^{m - 1} + 2^{3m} . 3^m \]
\[ = 7^2 \left( 7^{2m} + 2^{3m - 3} . 3^{m - 1} \right) + 2^{3m} . 3^m \left( 1 - \frac{49}{24} \right)\]
\[ = 7^2 \times 25\lambda - 2^{3m} . 3^m \times \frac{25}{2^3 . 3^1} \left[ \text{ Using } (1) \right]\]
\[ = 25\left( 49\lambda - 2^{3m - 3} . 3^{m - 1} \right) \]
\[\text{ It is divisible by } 25 . \]
\[\text{ Thus, } P\left( m + 1 \right) \text{ is true .} \]
\[\text{ By the principle of mathematical induction, P(n) is true for all n } \in N .\]
APPEARS IN
संबंधित प्रश्न
Prove the following by using the principle of mathematical induction for all n ∈ N:
Prove the following by using the principle of mathematical induction for all n ∈ N:
Prove the following by using the principle of mathematical induction for all n ∈ N:
Prove the following by using the principle of mathematical induction for all n ∈ N:
Prove the following by using the principle of mathematical induction for all n ∈ N:
Prove the following by using the principle of mathematical induction for all n ∈ N: `1+2+ 3+...+n<1/8(2n +1)^2`
Prove the following by using the principle of mathematical induction for all n ∈ N: n (n + 1) (n + 5) is a multiple of 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.
Prove the following by using the principle of mathematical induction for all n ∈ N (2n +7) < (n + 3)2
If P (n) is the statement "n(n + 1) is even", then what is P(3)?
1 + 3 + 5 + ... + (2n − 1) = n2 i.e., the sum of first n odd natural numbers is n2.
\[\frac{1}{2 . 5} + \frac{1}{5 . 8} + \frac{1}{8 . 11} + . . . + \frac{1}{(3n - 1)(3n + 2)} = \frac{n}{6n + 4}\]
1.3 + 3.5 + 5.7 + ... + (2n − 1) (2n + 1) =\[\frac{n(4 n^2 + 6n - 1)}{3}\]
1.2 + 2.3 + 3.4 + ... + n (n + 1) = \[\frac{n(n + 1)(n + 2)}{3}\]
2.7n + 3.5n − 5 is divisible by 24 for all n ∈ N.
11n+2 + 122n+1 is divisible by 133 for all n ∈ N.
\[\frac{(2n)!}{2^{2n} (n! )^2} \leq \frac{1}{\sqrt{3n + 1}}\] for all n ∈ N .
Prove that the number of subsets of a set containing n distinct elements is 2n, for all n \[\in\] N .
\[\text{ A sequence } a_1 , a_2 , a_3 , . . . \text{ is defined by letting } a_1 = 3 \text{ and } a_k = 7 a_{k - 1} \text{ for all natural numbers } k \geq 2 . \text{ Show that } a_n = 3 \cdot 7^{n - 1} \text{ for all } n \in N .\]
Prove by method of induction, for all n ∈ N:
1.3 + 3.5 + 5.7 + ..... to n terms = `"n"/3(4"n"^2 + 6"n" - 1)`
Prove by method of induction, for all n ∈ N:
5 + 52 + 53 + .... + 5n = `5/4(5^"n" - 1)`
Answer the following:
Prove by method of induction loga xn = n logax, x > 0, n ∈ N
The distributive law from algebra says that for all real numbers c, a1 and a2, we have c(a1 + a2) = ca1 + ca2.
Use this law and mathematical induction to prove that, for all natural numbers, n ≥ 2, if c, a1, a2, ..., an are any real numbers, then c(a1 + a2 + ... + an) = ca1 + ca2 + ... + can.
Prove the statement by using the Principle of Mathematical Induction:
4n – 1 is divisible by 3, for each natural number n.
Prove the statement by using the Principle of Mathematical Induction:
23n – 1 is divisible by 7, for all natural numbers n.
Prove the statement by using the Principle of Mathematical Induction:
32n – 1 is divisible by 8, for all natural numbers n.
Prove the statement by using the Principle of Mathematical Induction:
n2 < 2n for all natural numbers n ≥ 5.
Prove the statement by using the Principle of Mathematical Induction:
2n < (n + 2)! for all natural number n.
Prove the statement by using the Principle of Mathematical Induction:
2 + 4 + 6 + ... + 2n = n2 + n for all natural numbers n.
Prove the statement by using the Principle of Mathematical Induction:
1 + 2 + 22 + ... + 2n = 2n+1 – 1 for all natural numbers n.
A sequence d1, d2, d3 ... is defined by letting d1 = 2 and dk = `(d_(k - 1))/"k"` for all natural numbers, k ≥ 2. Show that dn = `2/(n!)` for all n ∈ N.
Prove that `1/(n + 1) + 1/(n + 2) + ... + 1/(2n) > 13/24`, for all natural numbers n > 1.
If P(n): 2n < n!, n ∈ N, then P(n) is true for all n ≥ ______.
State whether the following statement is true or false. Justify.
Let P(n) be a statement and let P(k) ⇒ P(k + 1), for some natural number k, then P(n) is true for all n ∈ N.
