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Hi, I don't understand how to do this question. Can someone please help and explain how to do it?

 

In the expansion of (1+x)n, three consecutive coefficients are in the ratio 1:7:35. Find the positive integer n.

 Jul 29, 2023
 #1
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Let the 3 consecutive co-efficients be 

 

     (nr1),(nr),(nr+1)And these are in the ratio 1:7:35 so(nr1)=17(nr)(1)and(nr+1)=5(nr)(2) 

(nr1)=17(nr)(1) LHS=n!(r1)!(nr+1)! LHS=n!r!(r)(nr)!(nr+1) LHS=n!(r)r!(nr)!(nr+1) LHS=(nr)(r)(nr+1) so17=rnr+1nr+1=7rn=8r1

 

Now simplify equation 2 and then solve simulataneously to find n and r.     Then check your answer

 

 

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LaTex:

\binom{n}{r-1},\qquad \binom{n}{r},\qquad \binom{n}{r+1}\\
\text{And these are in the ratio  }\qquad 1:7:35\\~\\
so\\
\binom{n}{r-1}=\frac{1}{7}\cdot \binom{n}{r}\qquad \color{red}{(1)} \color{black}{\quad and } \qquad 
 \binom{n}{r+1}=5\cdot \binom{n}{r}\quad \color{red}{(2)}\\~\\

 

\binom{n}{r-1}=\frac{1}{7}\cdot \binom{n}{r}\qquad \color{red}{(1)} \\~\\
LHS=\frac{n!}{(r-1)!(n-r+1)!}\\~\\
LHS=\frac{n!}{\frac{r!}{(r)}(n-r)!(n-r+1)}\\~\\
LHS=\frac{n!\qquad (r)}{r!(n-r)!\quad (n-r+1)}\\~\\
LHS=\binom{n}{r}\frac{ (r)}{ (n-r+1)}\\~\\
so\\
\frac{1}{7}=\frac{r}{n-r+1}\\
n-r+1=7r\\
n=8r-1

 Jul 30, 2023

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