Question

Which term has greatest coefficient in the expansion of $(1+2x/3{)}^{15}?$

• A. $5^{\text{th }}$
• B. $6^{\text{th }}$
• C. $7^{\text{th }}$
• D. $8^{\text{th }}$

Answer 1

Answer: (A)

The greatest coefficient is equal to the greatest term when $x=1$.
For $x=1,\frac{T_{r+1}}{T_r}=\frac{15-r+1}{r}\times \frac{2}{3}$
Let $\frac{T_{r+1}}{T_r}\ge 1$
or $\frac{15-r+1}{r}\frac{2}{3}\ge 1$
or $32-2r\ge 3r$
or $r\le 32/5$
or $r=6$
Hence, $7^{\text{th }}$ term has the greatest coefficient and its value is $T_{6+1}={}^{15}{C}_6(2/3{)}^6$