Advertisements
Advertisements
प्रश्न
By the principle of mathematical induction, prove the following:
52n – 1 is divisible by 24, for all n ∈ N.
उत्तर
Let P(n) be the proposition that 52n – 1 is divisible by 24.
For n = 1, P(1) is: 52 – 1 = 25 – 1 = 24, 24 is divisible by 24.
Assume that P(k) is true.
i.e., 52k – 1 is divisible by 24
Let 52k – 1 = 24m
To prove P(k + 1) is true.
i.e., to prove `5^(2(k+1)) - 1` is divisible by 24.
P(k): 52k – 1 is divisible by 24.
P(k + 1) = `5^(2(k+1)) - 1`
= 52k . 52 – 1
= 52k (25) – 1
= 52k (24 + 1) – 1
= 24 . 52k + 52k – 1
= 24 . 52k + 24m
= 24 [52k + 24]
which is divisible by 24 ⇒ P(k + 1) is also true.
Hence by mathematical induction, P(n) is true for all values n ∈ N.
APPEARS IN
संबंधित प्रश्न
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:
2n > n, for all n ∈ N.
The term containing x3 in the expansion of (x – 2y)7 is:
Prove that the sum of the first n non-zero even numbers is n2 + n
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
By the principle of Mathematical induction, prove that, for n ≥ 1
`1^2 + 2^2 + 3^2 + ... + "n"^2 > "n"^2/3`
Choose the correct alternative:
In 3 fingers, the number of ways four rings can be worn is · · · · · · · · · ways
