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in Mathematical Induction by (44.1k points)
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Prove by the method of induction, for all n ∈ N.

\(\frac1{3.4.5}+\frac2{4.5.6}+\frac3{5.6.7}+....+\frac{n}{(n+2)(n+3)(n+4)}\) \(=\frac{n(n+1)}{6(n+3)(n+4)}\)

1/3.4.5 + 2/4.5.6 + 3/5.6.7 + .... + n/((n + 2)(n + 3)(n + 4))

1 Answer

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Let P(n) =  \(\frac1{3.4.5}+\frac2{4.5.6}+\frac3{5.6.7}+....+\frac{n}{(n+2)(n+3)(n+4)}\) \(=\frac{n(n+1)}{6(n+3)(n+4)}\), for all n ∈ N.

Step I:

Step II:

Let us assume that P(n) is true for n = k.

Step III:

We have to prove that P(n) is true for n = k + 1,

i.e., to prove that

Step IV:

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

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