A) 4
B) 5
C) 6
D) 7
Correct Answer: B
Solution :
| [b] \[{{A}_{n}}=704+\frac{704}{2}+\frac{704}{4}+....\] to n terms |
| \[=\frac{704\left( 1-{{\left( \frac{1}{2} \right)}^{n}} \right)}{1-\frac{1}{2}}=704\times 2\left( 1-{{\left( \frac{1}{2} \right)}^{n}} \right)\] |
| \[{{B}_{n}}=1984-\frac{1984}{2}+\frac{1984}{4}....\] to n terms |
| \[=\frac{1984\left( 1-{{\left( \frac{-1}{2} \right)}^{n}} \right)}{1-\left( \frac{-1}{2} \right)}=1984\times \frac{2}{3}\left( 1-{{\left( \frac{-1}{2} \right)}^{n}} \right)\] |
| Now, \[{{A}_{n}}={{B}_{n}}\Rightarrow 704\times 2\left( 1-{{\left( \frac{1}{2} \right)}^{n}} \right)\] |
| \[=1984\times \frac{2}{3}\times \left( 1-{{\left( \frac{-1}{2} \right)}^{n}} \right)\] |
| \[\Rightarrow 33-31=33{{\left( \frac{1}{2} \right)}^{n}}-31{{\left( \frac{-1}{2} \right)}^{n}}\] |
| \[\Rightarrow {{2}^{n+1}}=33-31{{(-1)}^{n}}\Rightarrow n=5\] |
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