Question

If the coefficient of the three successive terms in the binomial expansion of \((1+x)^n\) are in the ratio \(1:7:42\), then the first of these terms in the expansion is

• A. \(9^{\text{th}}\)
• B. \(6^{\text{th}}\)
• C. \(8^{\text{th}}\)
• D. \(7^{\text{th}}\)

Hint:

Use the ratio of consecutive binomial coefficients, \(\frac{\binom nr}{\binom n{r-1}}=\frac{n-r+1}{r}\), twice.

Answer 1

The Correct Option is D

Let the three coefficients be \(\binom{n}{r-1},\binom{n}{r},\binom{n}{r+1}\) in the ratio \(1:7:42\).
\(\frac{\binom nr}{\binom n{r-1}}=\frac{n-r+1}{r}=7\Rightarrow n+1=8r\).
\(\frac{\binom n{r+1}}{\binom nr}=\frac{n-r}{r+1}=6\Rightarrow n=7r+6\).
Solving: \(8r-1=7r+6\Rightarrow r=7,\ n=55\).
The first coefficient is \(\binom{55}{r-1}=\binom{55}{6}\), the coefficient of \(x^6\), i.e. the \(7^{\text{th}}\) term.
Answer: (4) \(7^{\text{th}}\)