Advertisements
Advertisements
Question
Using the Mathematical induction, show that for any natural number n ≥ 2,
`(1 - 1/2^2)(1 - 1/3^2)(1 - 1/4^2) ... (1 - 1/"n"^2) = ("n" + 1)/2`
Advertisements
Solution
Let P(n) = is the statement `(1 -1/2^2)(1 -1/3^2) .... (1 - 1/"n"^2) = ("n" + 1)/2`
Given n ≥ 2
For n = 2
L.H.S = `(1 - 1/2^2)`
= `1 - 1/4`
= `3/4`
R.H.S P(2) = `(2 + 1)/(2(2))`
= `3/4`
L.H.S = R.H.S
P(n) is true for n = 2
Assume tat P(n) is true for n = k
(i.e.) `(1 -1/2^2)(1 -1/3^2) .... (1 - 1/"k"^2) = ("k" + 1)/2` is true
To prove P(k + 1) is true
Now P(k + 1) = `"P"("k") xx("t"_("k" + 1))`
= `("k" + 1)/(2"k") xx (1 - 1/("k" + 1)^2)`
= `("k" + 1)/(2"k") xx [(("k" + 1)^2 - 1)/("k" + 1)^2]`
= `("k" + 1)/(2"k") xx ("k"^2 + 2"k" + 1 - 1)/("k" + 1)^2`
= `("k" + 1)/(2"k") xx ("k"("k" + 2))/("k" + 1)^2`
= `("k" + 2)/(2("k" + 1))`
= `(("k" + 1 + 1))/(2("k" + 1))`
⇒ P(k + 1) is true when P(k) is true so by the principle of mathematical induction P(n) is true.
APPEARS IN
RELATED QUESTIONS
By the principle of mathematical induction, prove the following:
1.2 + 2.3 + 3.4 + … + n(n + 1) = `(n(n + 1)(n + 2))/3` for all n ∈ N.
By the principle of mathematical induction, prove the following:
4 + 8 + 12 + ……. + 4n = 2n(n + 1), for all n ∈ N.
By the principle of mathematical induction, prove the following:
1 + 4 + 7 + ……. + (3n – 2) = `("n"(3"n" - 1))/2` for all n ∈ N.
By the principle of mathematical induction, prove the following:
32n – 1 is divisible by 8, for all n ∈ N.
By the principle of mathematical induction, prove the following:
52n – 1 is divisible by 24, for all n ∈ N.
By the principle of mathematical induction, prove the following:
n(n + 1) (n + 2) is divisible by 6, for all n ∈ N.
By the principle of mathematical induction, prove the following:
2n > n, for all n ∈ N.
The term containing x3 in the expansion of (x – 2y)7 is:
Using the Mathematical induction, show that for any natural number n,
`1/(1*2*3) + 1/(2*3*4) + 1/(3*4*5) + ... + 1/("n"("n" + 1)*("n" + 2)) = ("n"("n" + 3))/(4("n" + 1)("n" + 2))`
Using the Mathematical induction, show that for any natural number n,
`1/(2.5) + 1/(5.8) + 1/(8.11) + ... + 1/((3"n" - 1)(3"n" + 2)) = "n"/(6"n" + 4)`
Prove by Mathematical Induction that
1! + (2 × 2!) + (3 × 3!) + ... + (n × n!) = (n + 1)! − 1
Using the Mathematical induction, show that for any natural number n, x2n − y2n is divisible by x + y
By the principle of Mathematical induction, prove that, for n ≥ 1
`1^2 + 2^2 + 3^2 + ... + "n"^2 > "n"^2/3`
Use induction to prove that n3 − 7n + 3, is divisible by 3, for all natural numbers n
Prove that using the Mathematical induction
`sin(alpha) + sin (alpha + pi/6) + sin(alpha + (2pi)/6) + ... + sin(alpha + (("n" - 1)pi)/6) = (sin(alpha + (("n" - 1)pi)/12) xx sin(("n"pi)/12))/(sin (pi/12)`
Choose the correct alternative:
Everybody in a room shakes hands with everybody else. The total number of shake hands is 66. The number of persons in the room is ______
