Advertisements
Advertisements
प्रश्न
\[\frac{(2n)!}{2^{2n} (n! )^2} \leq \frac{1}{\sqrt{3n + 1}}\] for all n ∈ N .
उत्तर
Let P(n) be the given statement.
Thus, we have .
\[P\left( n \right): \frac{\left( 2n \right)!}{2^{2n} \left( n! \right)^2} \leq \frac{1}{\sqrt{3n + 1}}\]
\[\text{ Step1} : \]
\[P(1): \frac{2!}{2^2 . 1} = \frac{1}{2} \leq \frac{1}{\sqrt{3 + 1}}\]
\[\text{ Thus, P(1) is true} . \]
\[\text{ Step2: } \]
\[\text{ Let P(m) be true .} \]
\[\text{ Thus, we have: } \]
\[\frac{\left( 2m \right)!}{2^{2m} \left( m! \right)^2} \leq \frac{1}{\sqrt{3m + 1}}\]
\[\text{ We need to prove that P(m + 1) is true .} \]
Now,
\[P(m + 1): \]
\[\frac{(2m + 2)!}{2^{2m + 2} \left( (m + 1)! \right)^2} = \frac{\left( 2m + 2 \right)\left( 2m + 1 \right)\left( 2m \right)!}{2^{2m} . 2^2 \left( m + 1 \right)^2 \left( m! \right)^2}\]
\[ \Rightarrow \frac{(2m + 2)!}{2^{2m + 2} \left( (m + 1)! \right)^2} \leq \frac{\left( 2m \right)!}{2^{2m} \left( m! \right)^2} \times \frac{\left( 2m + 2 \right)\left( 2m + 1 \right)}{2^2 \left( m + 1 \right)^2}\]
\[ \Rightarrow \frac{(2m + 2)!}{2^{2m + 2} \left( (m + 1)! \right)^2} \leq \frac{2m + 1}{2\left( m + 1 \right)\sqrt{3m + 1}}\]
\[\Rightarrow \frac{\left( 2m + 2 \right)!}{2^{2m + 2} \left( \left( m + 1 \right)! \right)^2} \leq \sqrt{\frac{\left( 2m + 1 \right)^2}{4 \left( m + 1 \right)^2 \left( 3m + 1 \right)}}\]
\[ \Rightarrow \frac{\left( 2m + 2 \right)!}{2^{2m + 2} \left( \left( m + 1 \right)! \right)^2} \leq \sqrt{\frac{\left( 4 m^2 + 4m + 1 \right) \times \left( 3m + 4 \right)}{4\left( 3 m^3 + 7 m^2 + 5m + 1 \right)\left( 3m + 4 \right)}}\]
\[ \Rightarrow \frac{\left( 2m + 2 \right)!}{2^{2m + 2} \left( \left( m + 1 \right)! \right)^2} \leq \sqrt{\frac{12 m^3 + 28 m^2 + 19m + 4}{\left( 12 m^3 + 28 m^2 + 20m + 4 \right)\left( 3m + 4 \right)}}\]
\[ \because \frac{12 m^3 + 28 m^2 + 19m + 4}{\left( 12 m^3 + 28 m^2 + 20m + 4 \right)} < 1\]
\[ \therefore \frac{\left( 2m + 2 \right)!}{2^{2m + 2} \left( \left( m + 1 \right)! \right)^2} < \frac{1}{\sqrt{3m + 4}}\]
Thus, P(m + 1) is true.
Hence, by mathematical induction
\[\frac{(2n)!}{2^{2n} (n! )^2} \leq \frac{1}{\sqrt{3n + 1}}\] is true for all n ∈ N
APPEARS IN
संबंधित प्रश्न
Prove the following by using the principle of mathematical induction for all n ∈ N:
`1^3 + 2^3 + 3^3 + ... + n^3 = ((n(n+1))/2)^2`
Prove the following by using the principle of mathematical induction for all n ∈ N:
1.2 + 2.3 + 3.4+ ... + n(n+1) = `[(n(n+1)(n+2))/3]`
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: 32n + 2 – 8n– 9 is divisible by 8.
If P (n) is the statement "n3 + n is divisible by 3", prove that P (3) is true but P (4) is not true.
If P (n) is the statement "n2 + n is even", and if P (r) is true, then P (r + 1) is true.
If P (n) is the statement "n2 − n + 41 is prime", prove that P (1), P (2) and P (3) are true. Prove also that P (41) is not true.
1 + 3 + 32 + ... + 3n−1 = \[\frac{3^n - 1}{2}\]
\[\frac{1}{1 . 2} + \frac{1}{2 . 3} + \frac{1}{3 . 4} + . . . + \frac{1}{n(n + 1)} = \frac{n}{n + 1}\]
\[\frac{1}{1 . 4} + \frac{1}{4 . 7} + \frac{1}{7 . 10} + . . . + \frac{1}{(3n - 2)(3n + 1)} = \frac{n}{3n + 1}\]
\[\frac{1}{3 . 7} + \frac{1}{7 . 11} + \frac{1}{11 . 5} + . . . + \frac{1}{(4n - 1)(4n + 3)} = \frac{n}{3(4n + 3)}\]
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) =\[\frac{n(4 n^2 + 6n - 1)}{3}\]
1.2 + 2.3 + 3.4 + ... + n (n + 1) = \[\frac{n(n + 1)(n + 2)}{3}\]
a + ar + ar2 + ... + arn−1 = \[a\left( \frac{r^n - 1}{r - 1} \right), r \neq 1\]
x2n−1 + y2n−1 is divisible by x + y for all n ∈ 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:
12 + 32 + 52 + .... + (2n − 1)2 = `"n"/3 (2"n" − 1)(2"n" + 1)`
Prove by method of induction, for all n ∈ N:
`1/(1.3) + 1/(3.5) + 1/(5.7) + ... + 1/((2"n" - 1)(2"n" + 1)) = "n"/(2"n" + 1)`
Prove by method of induction, for all n ∈ N:
Given that tn+1 = 5tn + 4, t1 = 4, prove that tn = 5n − 1
Answer the following:
Prove, by method of induction, for all n ∈ N
12 + 42 + 72 + ... + (3n − 2)2 = `"n"/2 (6"n"^2 - 3"n" - 1)`
Prove statement by using the Principle of Mathematical Induction for all n ∈ N, that:
1 + 3 + 5 + ... + (2n – 1) = n2
Prove statement by using the Principle of Mathematical Induction for all n ∈ N, that:
`sum_(t = 1)^(n - 1) t(t + 1) = (n(n - 1)(n + 1))/3`, for all natural numbers n ≥ 2.
Prove statement by using the Principle of Mathematical Induction for all n ∈ N, that:
22n – 1 is divisible by 3.
A student was asked to prove a statement P(n) by induction. He proved that P(k + 1) is true whenever P(k) is true for all k > 5 ∈ N and also that P(5) is true. On the basis of this he could conclude that P(n) is true ______.
Give an example of a statement P(n) which is true for all n ≥ 4 but P(1), P(2) and P(3) are not true. Justify your answer
Prove the statement by using the Principle of Mathematical Induction:
For any natural number n, xn – yn is divisible by x – y, where x and y are any integers with x ≠ y.
Prove the statement by using the Principle of Mathematical Induction:
n(n2 + 5) is divisible by 6, for each natural number n.
Prove the statement by using the Principle of Mathematical Induction:
`sqrt(n) < 1/sqrt(1) + 1/sqrt(2) + ... + 1/sqrt(n)`, for all natural numbers n ≥ 2.
Prove the statement by using the Principle of Mathematical Induction:
2 + 4 + 6 + ... + 2n = n2 + n 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 10n + 3.4n+2 + k is divisible by 9 for all n ∈ N, then the least positive integral value of k is ______.
If xn – 1 is divisible by x – k, then the least positive integral value of k is ______.
Consider the statement: “P(n) : n2 – n + 41 is prime." Then which one of the following is true?
