Question

The coefficients of 5th, 6th and 7th terms in the expansion of (1 + x)ⁿ are in A.P., find n.

Answer 1

$$\begin{gathered} \mathrm{Coefficients}\mathrm{of}\mathrm{the}5\mathrm{th},6\mathrm{th}\mathrm{and}7\mathrm{th}\mathrm{terms}\mathrm{in}\mathrm{the}\mathrm{given}\mathrm{expansion}\mathrm{are}{}^{n}C_4,{}^{n}C_5\mathrm{and}{}^{n}C_6 \\ \mathrm{These}\mathrm{coefficients}\mathrm{are}\mathrm{in}AP. \\ \mathrm{Thus},\mathrm{we}\mathrm{have} \\ 2{}^{n}C_5={}^{n}C_4+{}^{n}C_6 \\ \mathrm{On}\mathrm{dividing}\mathrm{both}\mathrm{sides}\mathrm{by}{}^{n}C_5,weget: \\ 2=\frac{{}^{n}C_4}{{}^{n}C_5}+\frac{{}^{n}C_6}{{}^{n}C_5} \\ \Rightarrow 2=\frac{5}{n-4}+\frac{n-5}{6} \\ \Rightarrow 12n-48=30+n^2-4n-5n+20 \\ \Rightarrow n^2-21n+98=0 \\ \Rightarrow (n-14)(n-7)=0 \\ \Rightarrow n=7\mathrm{or}14 \end{gathered}$$