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Exercise 4.1 · Q16

Q.Find the value of nn if

(i) (n+1)!=20(n−1)!(n+1)! = 20(n-1)!
(ii) 18!+19!=n10!\dfrac{1}{8!}+\dfrac{1}{9!} = \dfrac{n}{10!}.
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Rewrite every factorial as a multiple of the smallest one appearing, then equate.

Step 1. (i) (n+1)!=20(n−1)!⇒(n+1)n(n−1)!=20(n−1)!⇒n(n+1)=20⇒n2+n−20=0⇒(n+5)(n−4)=0(n+1)!=20(n-1)!\Rightarrow (n+1)n(n-1)!=20(n-1)!\Rightarrow n(n+1)=20\Rightarrow n^2+n-20=0\Rightarrow(n+5)(n-4)=0. Since nn is a positive integer, n=4n=4. …

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