(b)
For some positive integer \(n\) the expansion of \((1 + x)^n\)
has three consecutive coefficients \(a,~b,~c\) that satisfy
\(a:b:c = 1:7:35\).
What must \(n\) be?
\(\text{Let $a=\dbinom{n}{k - 1}$ } \\ \text{Let $b=\dbinom{n}{k}$ } \\ \text{Let $c=\dbinom{n}{k+1}$ }\)
\(\begin{array}{|rcll|} \hline \dfrac{b}{a} = \dfrac{7}{1} &=& \dfrac{\dbinom{n}{k}}{\dbinom{n}{k - 1}} \\\\ \dfrac{7}{1} &=& \dfrac{\dbinom{n}{k}}{\dbinom{n}{k - 1}} \\\\ \dfrac{7}{1} &=& \dfrac{n-k+1}{k} \\\\ 7k &=& n-k+1 \\ 8k &=& n+1 \\ \mathbf{n} &=& \mathbf{8k-1} \\ \hline \end{array}\)
\(\begin{array}{|rcll|} \hline \dbinom{n}{k + 1} &=& \dfrac{n!}{(k+1)!\Big( n-(k+1)\Big)!} \\\\ &=& \dfrac{n!}{(k+1)!\Big( n-k-1)\Big)!} \\\\ && \boxed{ (k+1)!=k!(k+1)} \\\\ &=& \dfrac{n!}{k!(k+1)( n-k-1)!} \\\\ && \boxed{ ( n-k-1)!(n-k)=(n-k)! \\ ( n-k-1)!=\dfrac{(n-k)!}{n-k} } \\\\ &=& \dfrac{n!*(n-k)}{k!(k+1)(n-k)!} \\\\ &=& \dfrac{n!}{k!(n-k)! } *\dfrac{(n-k)}{(k+1)} \\\\ && \boxed{ \dbinom{n}{k}=\dfrac{n!}{k!(n-k)! } } \\\\ &=& \dbinom{n}{k} *\dfrac{(n-k)}{(k+1)} \\ \hline \end{array}\)
\(\begin{array}{|rcll|} \hline \dfrac{c}{b} = \dfrac{35}{7} &=& \dfrac{\dbinom{n}{k+1}}{\dbinom{n}{k}} \\\\ \dfrac{35}{7} &=& \dfrac{\dbinom{n}{k} *\dfrac{(n-k)}{(k+1)}}{\dbinom{n}{k}} \\\\ \dfrac{35}{7} &=&\dfrac{(n-k)}{(k+1)} \\\\ 35(k+1) &=&7(n-k) \\ 42k &=& 7n-35 \\ \mathbf{k} &=& \mathbf{ \dfrac{7n-35}{42}} \\ \hline \end{array}\)
\(\begin{array}{|rcll|} \hline n &=& 8k-1 \quad | \quad \mathbf{k=\dfrac{7n-35}{42}} \\\\ n &=& 8\left(\dfrac{7n-35}{42}\right)-1 \\\\ n &=& \dfrac{4(7n-35)}{21}-1 \\\\ n &=& \dfrac{4(7n-35)-21}{21} \\\\ 21n &=&4(7n-35)-21\\ 21n &=&28n-140-21\\ 7n &=& 161 \quad | \quad :7 \\ \mathbf{n} &=& \mathbf{23} \\ \hline k &=& \dfrac{7n-35}{42} \\\\ k &=& \dfrac{7*23-35}{42} \\\\ \mathbf{k} &=& \mathbf{3} \\ \hline \end{array}\)
\(\begin{array}{|rcll|} \hline a:b:c =\dbinom{n}{k - 1}:\dbinom{n}{k}:\dbinom{n}{k + 1} &=& 1:7:35 \\\\ \dbinom{n}{k - 1}:\dbinom{n}{k}:\dbinom{n}{k + 1} &=& 1:7:35 \\\\ \dbinom{23}{3 - 1}:\dbinom{23}{3}:\dbinom{23}{3 + 1} &=& 1:7:35 \\\\ \dbinom{23}{2}:\dbinom{23}{3}:\dbinom{23}{4} &=& 1:7:35 \\\\ 253:1771:8855 &=& 1:7:35 \\ \hline \end{array}\)
