Advertisements
Advertisements
Question
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)`
Advertisements
Solution
Let P(n) = `1/(2.5) + 1/(5.8) + 1/(8.11) + ... + 1/((3"n" - 1)(3"n" + 2)) = "n"/(6"n" + 4)`
For n = 1
P(1) = `1/((3 xx 1 - 1)(3 xx 1 + 2))`
= `1/((6 xx 1 + 4))`
⇒ `1/(2 xx 5) = 1/10`
⇒ `1/10 = 1/10`
∴ P(1) is true
Let P(n) be true for n = k
∴ P(k) = `1/(2.5) + 1/(5.8) + 1/(8.11) + ... + 1/((3"k" - 1)(3"k" + 2))`
= `"k"/(6"k" + 4)` ......(i)
For n = k + 1
P(k + 1) = `1/(2.5) + 1/(5.8) + 1/(8.11) + ... + 1/((3"k" - 1)(3"k" + 2)) + 1/([(3"k" - 1) - 1][3("k" + 1) + 2])`
= `("k" + 1)/(6"k" + 10)`
= `"k"/(6"k" + 4) + 1/((3"kk" + 2)(3"k" + 5))`
= `1/((3"k" + 2))["k"/2 + 1/(3"k" + 5)]`
= `1/((3"k" + 2))[(3"k"^2 + 5"k" + 2)/(2(3"k" + 5))]`
= `1/((3"k" + 2)) [(3"k"^2 + 3"k" + 2"k" + 2)/(2(3"k" + 5))]`
= `1/((3"l" + 2))[(3"k"("k" + 1) + 2("k" + 1))/(2(3"k" + 5))]`
= `1/((3"k" + 2))[(("k" + 1)(3"k" + 2))/(2(3"k" + 5))]`
= `("k" + 1)/(6"k" + 10)`
∴ 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:
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:
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`
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`
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 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
Choose the correct alternative:
In 3 fingers, the number of ways four rings can be worn is · · · · · · · · · ways
