Advertisements
Advertisements
प्रश्न
By the principle of mathematical induction, prove that, for n ≥ 1
13 + 23 + 33 + ... + n3 = `(("n"("n" + 1))/2)^2`
Advertisements
उत्तर
P(n) = 13 + 23 + 33 + ... + n3 = `(("n"("n" + 1))/2)^2`
For n = 1
P(1) = 1
= `[(1(1 + 1))/2]^2`
⇒ 1 = 1
∴ P(1) is true
Let P(n) be true for n = k
∴ P(k) = 13 + 23 + 33 + ... + k3
= `[("k"("k" + 1))/2]^2` ......(i)
From n = k + 1
P(k + 1) = 13 + 23 + 33 + ... + k3 + (k + 1)3
= `[("k"("k" + 1))/2]^2 + ("k" + 1)^3` ......[Using (i)]
= `("k" + 1)^2 ["k"^2/4 + "k" + 1]`
= `("k" + 1)^2 [("k"^2 + 4"k" + 4)/4]`
=`(("k" + 1)^2("k" + 2)^2)/4`
= `[(("k" + 1)("k" +2))/2]^2`
∴ P(k+ 1) is true.
Thus P(k) is true
⇒ (k + 1) is true.
Hence by principle of mathematical induction
P(n) is true for all n ∈ N.
APPEARS IN
संबंधित प्रश्न
By the principle of mathematical induction, prove the following:
13 + 23 + 33 + ….. + n3 = `("n"^2("n + 1")^2)/4` for all x ∈ 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:
an – bn is divisible by a – b, 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:
2n > n, for all n ∈ N.
By the principle of mathematical induction, prove that, for n ≥ 1
12 + 32 + 52 + ... + (2n − 1)2 = `("n"(2"n" - 1)(2"n" + 1))/3`
Prove that the sum of the first n non-zero even numbers is n2 + n
By the principle of Mathematical induction, prove that, for n ≥ 1
1.2 + 2.3 + 3.4 + ... + n.(n + 1) = `("n"("n" + 1)("n" + 2))/3`
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`
Using the Mathematical induction, show that for any natural number n ≥ 2,
`1/(1 + 2) + 1/(1 + 2 + 3) + 1/(1 +2 + 3 + 4) + .... + 1/(1 + 2 + 3 + ... + "n") = ("n" - 1)/("n" + 1)`
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
Use induction to prove that n3 − 7n + 3, is divisible by 3, for all natural numbers n
Use induction to prove that 5n+1 + 4 × 6n when divided by 20 leaves a remainder 9, for all natural numbers n
Use induction to prove that 10n + 3 × 4n+2 + 5, is divisible by 9, 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)`
