Advertisements
Advertisements
Question
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`
Solution
Let P(n) = 1.2 + 2.3 + 3.4 + ... + n.(n + 1) = `("n"("n" + 1)("n" + 2))/3`
For n = 1
P(1) = 1(1 + 1)
= `((1 + 1)(1 + 2))/3`
⇒ 2 = 2
∴ P(1) is true
Let P(n) b true for n = k
∴ P(k) = 1.2 + 2.3 + 3.4 +... + k(k + 1)
= `[("k"("k" + 1)("k" + 2))/3]` ......(i)
For n = k + 1
P(k + 1) = 1.2 + 2.3 + 3.4 .... + k(k + 1) + (k + 1)(k + 2)
= `(""("k" + 1)("k" + 2))/3 + ("k" + )("k"+ 2)` .....[Using (i)]
= `("k" + 1)("k" + )["k"/3 + 1]`
= `("k" + 1)("k" + 2)[(("k" + 3))/3]`
=`(("k" + 1)("k" + 2)("k" + 3))/3`
∴ P(k + 1) is true
Thus P(k) is true
⇒ P(k + 1) is true
Hence by principle of mathematical induction
P(n) is true for all n ∈ N
APPEARS IN
RELATED QUESTIONS
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:
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:
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:
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:
By the principle of mathematical induction, prove that, for n ≥ 1
13 + 23 + 33 + ... + n3 = `(("n"("n" + 1))/2)^2`
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`
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)`
Using the Mathematical induction, show that for any natural number n, x2n − y2n is divisible by x + y
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)`
Choose the correct alternative:
In 3 fingers, the number of ways four rings can be worn is · · · · · · · · · ways
Choose the correct alternative:
If `""^("a"^2 - "a")"C"_2 = ""^("a"^2 - "a")"C"_4` then the value of a is
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 ______
