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

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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

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Let the 3 consecutive co-efficients be

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

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

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

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LaTex:

\text{And these are in the ratio  }\qquad 1:7:35\\~\\
so\\