A) 7th and 8th
B) 8th and 9th
C) 28th and 29th
D) 27th and 28th
Correct Answer: A
Solution :
Let \[{{\text{r}}^{\text{th}}}\]and \[\text{(r+1)}{{\,}^{\text{th}}}\]term has equal coefficient\[{{\left( 2+\frac{x}{3} \right)}^{55}}={{2}^{55}}{{\left( 1+\frac{x}{6} \right)}^{55}}\] \[{{\text{r}}^{\text{th}}}\]term\[={{2}^{55}}{{\,}^{55}}{{C}_{r}}{{\left( \frac{x}{6} \right)}^{r}}\] Coefficient of \[{{x}^{r}}\] is\[{{2}^{55}}{{\,}^{55}}{{C}_{r}}\frac{1}{{{6}^{r}}}\] Coefficient of \[{{x}^{r+1}}\]is\[{{2}^{55}}{{\,}^{55}}{{C}_{r+1}}.\frac{1}{{{6}^{r+1}}}\] \[{{2}^{55}}{{\,}^{55}}{{C}_{r}}\frac{1}{{{6}^{r}}}={{2}^{55}}\,{{\,}^{55}}{{C}_{r+1}}\frac{1}{{{6}^{r+1}}}\] \[\frac{1}{\underline{|r|55-r}}=\frac{1}{\underline{|r+1|54-r}}.\frac{1}{6}\] \[6(r+1)=55-r\] \[6r+6=55-r\] \[7r=49\] \[r=7\] \[(r+1)=9\] Coefficient of 7th and 8th terms are equal.
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