Advertisements
Advertisements
प्रश्न
Advertisements
उत्तर
Let P(n) be the given statement.
\[\text{ Step } 1: \]
\[P(1): \sin x = \frac{\sin^2 x}{\sin x}\]
\[\text{ Thus, P(1) is true } . \]
\[\text{ Step 2: } \]
\[\text{ Let P(m) be true .} \]
\[ \therefore \sin x + \sin 3x + . . . + \sin\left( 2m - 1 \right)x = \frac{\sin^2 mx}{\sin x}\]
\[\text{ We shall show that P(m + 1) is true .} \]
\[\text{ We know that P(m) is true } . \]
\[ \therefore \sin x + \sin 3x + . . . + \sin (2m - 1) = \frac{\sin^2 mx}{\sin x}\]
\[ \Rightarrow \sin x + \sin 3x + . . . \sin (2m - 1)x + \sin (2m + 1)x = \frac{\sin^2 mx}{\sin x} + \sin (2m + 1)x \left( \text{ Adding } \sin (2m + 1)x \text{ to both the sides } \right)\]
\[ \Rightarrow P(m + 1)x = \frac{\sin^2 mx + \sin x\left[ \sin mx\cos\left( m + 1 \right)x + \sin\left( m + 1 \right)x \cos x \right]}{\sin x}\]
\[ = \frac{\sin^2 mx + \sin x\left( \sin mx\cos mxcos x - \sin^2 mx\sin x + \sin mx\cos x\cos mx + \cos^2 mx\sin x \right)}{\sin x}\]
\[ = \frac{\sin^2 mx + 2\sin x\cos x\cos mx - \sin^2 x \sin^2 mx + \cos^2 mx \sin^2 x}{\sin x}\]
\[ = \frac{\sin^2 mx\left( 1 - \sin^2 x \right) + 2\sin x\cos x\cos mx + \cos^2 mx \sin^2 x}{\sin x}\]
\[ = \frac{\sin^2 mx \cos^2 x + 2\sin x\cos x\cos mx + \cos^2 mx \sin^2 x}{\sin x}\]
\[ = \frac{\left( \sin mx \cos x + \cos mx \sin x \right)^2}{\sin x}\]
\[ = \frac{\left[ \sin\left( m + 1 \right) \right]^2}{\sin x}\]
\[\text{ [Hence, P(m + 1) is true } . \]
\[ \text{ By the principle of mathematical induction, the given statement P(n) is true for all } n \in N . \]
APPEARS IN
संबंधित प्रश्न
Prove the following by using the principle of mathematical induction for all n ∈ N:
`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 + 1/4 + 1/8 + ... + 1/2^n = 1 - 1/2^n`
Prove the following by using the principle of mathematical induction for all n ∈ N:
(1+3/1)(1+ 5/4)(1+7/9)...`(1 + ((2n + 1))/n^2) = (n + 1)^2`
Prove the following by using the principle of mathematical induction for all n ∈ N:
`(1+ 1/1)(1+ 1/2)(1+ 1/3)...(1+ 1/n) = (n + 1)`
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 "n(n + 1) is even", then what is P(3)?
Given an example of a statement P (n) such that it is true for all n ∈ N.
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 + 2 + 3 + ... + n = \[\frac{n(n + 1)}{2}\] i.e. the sum of the first n natural numbers is \[\frac{n(n + 1)}{2}\] .
12 + 22 + 32 + ... + n2 =\[\frac{n(n + 1)(2n + 1)}{6}\] .
1 + 3 + 32 + ... + 3n−1 = \[\frac{3^n - 1}{2}\]
\[\frac{1}{1 . 2} + \frac{1}{2 . 3} + \frac{1}{3 . 4} + . . . + \frac{1}{n(n + 1)} = \frac{n}{n + 1}\]
1.3 + 3.5 + 5.7 + ... + (2n − 1) (2n + 1) =\[\frac{n(4 n^2 + 6n - 1)}{3}\]
a + ar + ar2 + ... + arn−1 = \[a\left( \frac{r^n - 1}{r - 1} \right), r \neq 1\]
32n+7 is divisible by 8 for all n ∈ N.
2.7n + 3.5n − 5 is divisible by 24 for all n ∈ N.
Given \[a_1 = \frac{1}{2}\left( a_0 + \frac{A}{a_0} \right), a_2 = \frac{1}{2}\left( a_1 + \frac{A}{a_1} \right) \text{ and } a_{n + 1} = \frac{1}{2}\left( a_n + \frac{A}{a_n} \right)\] for n ≥ 2, where a > 0, A > 0.
Prove that \[\frac{a_n - \sqrt{A}}{a_n + \sqrt{A}} = \left( \frac{a_1 - \sqrt{A}}{a_1 + \sqrt{A}} \right) 2^{n - 1}\]
Prove that 1 + 2 + 22 + ... + 2n = 2n+1 - 1 for all n \[\in\] N .
x2n−1 + y2n−1 is divisible by x + y for all n ∈ N.
Prove that the number of subsets of a set containing n distinct elements is 2n, for all n \[\in\] N .
Prove by method of induction, for all n ∈ N:
2 + 4 + 6 + ..... + 2n = n (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:
`[(1, 2),(0, 1)]^"n" = [(1, 2"n"),(0, 1)]` ∀ n ∈ N
Answer the following:
Prove, by method of induction, for all n ∈ N
8 + 17 + 26 + … + (9n – 1) = `"n"/2(9"n" + 7)`
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)`
Answer the following:
Prove, by method of induction, for all n ∈ N
2 + 3.2 + 4.22 + ... + (n + 1)2n–1 = n.2n
Answer the following:
Prove by method of induction 152n–1 + 1 is divisible by 16, for all n ∈ N.
Answer the following:
Prove by method of induction 52n − 22n is divisible by 3, for all n ∈ N
Prove statement by using the Principle of Mathematical Induction for all n ∈ N, that:
`(1 - 1/2^2).(1 - 1/3^2)...(1 - 1/n^2) = (n + 1)/(2n)`, for all natural numbers, n ≥ 2.
Define the sequence a1, a2, a3 ... as follows:
a1 = 2, an = 5 an–1, for all natural numbers n ≥ 2.
Use the Principle of Mathematical Induction to show that the terms of the sequence satisfy the formula an = 2.5n–1 for all natural numbers.
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:
1 + 2 + 22 + ... + 2n = 2n+1 – 1 for all natural numbers n.
Prove the statement by using the Principle of Mathematical Induction:
1 + 5 + 9 + ... + (4n – 3) = n(2n – 1) for all natural numbers n.
A sequence b0, b1, b2 ... is defined by letting b0 = 5 and bk = 4 + bk – 1 for all natural numbers k. Show that bn = 5 + 4n for all natural number n using mathematical induction.
Prove that, cosθ cos2θ cos22θ ... cos2n–1θ = `(sin 2^n theta)/(2^n sin theta)`, for all n ∈ N.
For all n ∈ N, 3.52n+1 + 23n+1 is divisible by ______.
Consider the statement: “P(n) : n2 – n + 41 is prime." Then which one of the following is true?
