Answer the following: Prove, by method of induction, for all n ∈ N 13.4.5+24.5.6+35.6.7+...+n(n+2)(n+3)(n+4)=n(n+1)6(n+3)(n+4)

प्रश्न

Answer the following:

Prove, by method of induction, for all n ∈ N

`1/(3.4.5) + 2/(4.5.6) + 3/(5.6.7) + ... + "n"/(("n" + 2)("n" + 3)("n" + 4)) = ("n"("n" + 1))/(6("n" + 3)("n" + 4))`

बेरीज

उत्तर

Let P(n) ≡ `1/(3.4.5) + 2/(4.5.6) + 3/(5.6.7) + ... + "n"/(("n" + 2)("n" + 3)("n" + 4)) = ("n"("n" + 1))/(6("n" + 3)("n" + 4))`, for all n ∈ N.

Step 1:

For n = 1, L.H.S. = `1/(3.4.5) = 1/60`

R.H.S. = `(1(1+1))/(6(1+3)(1+4))=2/(6(4)(5))=1/60`

∴ L.H.S. = R.H.S. for n = 1.

∴ P(1) is true.

Step 2:

Let us assume that for some k ∈ N, P(k) is true,

i.e., `1/(3.4.5) + 2/(4.5.6) + 3/(5.6.7) + ... + "k"/(("k" + 2)("k" + 3)("k" + 4)) = ("k"("k" + 1))/(6("k" + 3)("k" + 4))` ...(1)

Step 3:

To prove that P(k + 1) is true, i.e., to prove that

`1/(3.4.5) + 2/(4.5.6) + 3/(5.6.7) + ... + "k"/(("k" + 2)("k" + 3)("k" + 4)) + ("k" + 1)/(("k" + 3)("k" + 4)("k" + 5)) = (("k" + 1)("k" + 2))/(6("k" + 4)("k" + 5))`

Now, L.H.S. = `1/(3.4.5) + 2/(4.5.6) + 3/(5.6.7) + ... + "k"/(("k" + 2)("k" + 3)("k" + 4)) + ("k" + 1)/(("k" + 3)("k" + 4)("k" + 5))`

= `("k"("k" + 1))/(6("k" + 3)("k" + 4)) + ("k" + 1)/(("k" + 3)("k" + 4)("k" + 5))` ...[By (1)]

= `("k" + 1)/(("k" + 3)("k" + 4))["k"/6 + 1/("k" + 5)]`

= `("k" + 1)/(("k" + 3)("k" + 4))[("k"^2 + 5"k" + 6)/(6("k" + 5))]`

= `("k" + 1)/(("k" + 3)("k" + 4)) xx (("k" + 2)("k" + 3))/(6("k" + 5))`

= `(("k" + 1)("k" + 2))/(6("k" + 4)("k" + 5))`

= R.H.S.

∴ P(k + 1) is true.

Step 4:

From all the above steps and by the principle of mathematical induction P(n) is true for all n ∈ N,

i.e., `1/(3.4.5) + 2/(4.5.6) + 3/(5.6.7) + ... + "n"/(("n" + 2)("n" + 3)("n" + 4)) = ("n"("n" + 1))/(6("n" + 3)("n" + 4))`, for all n ∈ N.

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Principle of Mathematical Induction

या प्रश्नात किंवा उत्तरात काही त्रुटी आहे का?

Q II. (1) (iv)Q II. (1) (iii)Q II. (2)

पाठ 4: Methods of Induction and Binomial Theorem - Miscellaneous Exercise 4.2 [पृष्ठ ८५]

APPEARS IN

बालभारती Mathematics and Statistics (Arts and Science) Part 2 [English] Standard 11 Maharashtra State Board

पाठ 4 Methods of Induction and Binomial Theorem
Miscellaneous Exercise 4.2 | Q II. (1) (iv) | पृष्ठ ८५

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