Advertisements
Advertisements
Question
Use induction to prove that n3 − 7n + 3, is divisible by 3, for all natural numbers n
Solution
Let P(n): n3 – 7n + 3
Step 1:
P(1) = (1)3 – 7(1) + 3
= 1 – 7 + 3
= – 3
Which is divisible by 3
So, it is true for P(1).
Step 2:
P(k): k3 – 7k + 3 = 3λ.
Let it be true
⇒ k3 = 3λ + 7k – 3
Step 3:
P(k + 1) = (k + 1)3 – 7(k + 1) + 3
= k3 + 1 + 3k2 + 3k – 7k – 7 + 3
= k3 + 3k2 – 4k – 3
= (3λ + 7k – 3) + 3k2 – 4k – 3 ......(From Step 2)
= 3k2 + 3k + 3λ – 6
= 3(k2 + k + λ – 2)
Which is divisible by 3.
So it is true for P(k + 1).
Hence, P(k + 1) is true whenever it is true for P(k).
APPEARS IN
RELATED QUESTIONS
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:
n(n + 1) (n + 2) is divisible by 6, 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`
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) + 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,
`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 10n + 3 × 4n+2 + 5, is divisible by 9, for all natural numbers n
Choose the correct alternative:
In 3 fingers, the number of ways four rings can be worn is · · · · · · · · · ways
Choose the correct alternative:
1 + 3 + 5 + 7 + · · · + 17 is equal to
