Question

can someone pls solve Q 17 & 19

Answer 1

17.  Coefficient of 2 r+ 4 th term  = coefficient of r - 2 th term  in the expansion of (1 + x)¹⁸

The coefficients in a binomial expansion are symmetric around the center term.  As n = 18, there are 19 terms in the expansion. So the coefficients in the 1st 9 terms match with the coefficients of the last 9 terms, in the reverse way.  10th term is the middle term.

So:      10 - (r - 2)   =  (2 r + 4) - 10
 =>      3 r = 18   => r = 6
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19.
Coefficient of r+1 th term in the expansion of (1 +x)ⁿ⁺¹ =  ${}^{n+1}C_{r+1}=\frac{(n+1)!}{(r+1)! (n-r)!}$
 
Coefficient of r th term + that of r+1 th term in the expansion of (1+x)ⁿ = 
${}^{n}C_r+{}^nC_{r+1}\\\\=\frac{n!}{r!\ (n-r)!}+\frac{n!}{(r+1)!\ (n-r-1)!}\\\\=\frac{n!}{(r+1)!\ (n-r)!} [r+1+n-r ]=\frac{(n+1)!}{(r+1)! \ [n+1-(r+1)]}\\\\={}^{n+1}C_{r+1}$

Proved.