Advertisements
Advertisements
प्रश्न
By the principle of mathematical induction, prove the following:
n(n + 1) (n + 2) is divisible by 6, for all n ∈ N.
Advertisements
उत्तर
P(n): n(n + 1) (n + 2) is divisible by 6.
P(1): 1 (2) (3) = 6 is divisible by 6
∴ P(1) is true.
Let us assume that P(k) is true for n = k
That is, k (k + 1) (k + 2) = 6m for some m
To prove P(k + 1) is true i.e. to prove (k + 1) (k + 2)(k + 3) is divisible by 6.
P(k + 1) = (k + 1) (k + 2) (k + 3)
= (k + 1)(k + 2)k + 3(k + 1)(k + 2)
= 6m + 3(k + 1)(k + 2)
In the second term either k + 1 or k + 2 will be even, whatever be the value of k.
Hence second term is also divisible by 6.
∴ P (k + 1) is also true whenever P(k) is true.
By Mathematical Induction P (n) is true for all values of 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:
1 + 4 + 7 + ……. + (3n – 2) = `("n"(3"n" - 1))/2` 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.
The term containing x3 in the expansion of (x – 2y)7 is:
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) + 1/(1 + 2 + 3) + 1/(1 +2 + 3 + 4) + .... + 1/(1 + 2 + 3 + ... + "n") = ("n" - 1)/("n" + 1)`
Choose the correct alternative:
In 3 fingers, the number of ways four rings can be worn is · · · · · · · · · ways
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 ______
Choose the correct alternative:
1 + 3 + 5 + 7 + · · · + 17 is equal to
