MCQ
The sum to infinity of the following series $2 + \frac{1}{2} + \frac{1}{3} + \frac{1}{{{2^2}}} + \frac{1}{{{3^2}}} + \frac{1}{{{2^3}}} + \frac{1}{{{3^3}}} + ........$, will be
- A$3$
- B$4$
- ✓$7/2$
- D$9/2$
$ = \left( {1 + \frac{1}{2} + \frac{1}{{{2^2}}} + \frac{1}{{{2^3}}} + ......\infty } \right)$
+ $\left( {1 + \frac{1}{3} + \frac{1}{{{3^2}}} + \frac{1}{{{3^3}}} + ......\infty } \right)$
$ = \left( {\frac{1}{{1 - (1/2)}}} \right) + \left( {\frac{1}{{1 - (1/3)}}} \right) = 2 + \frac{3}{2} = \frac{7}{2}$.
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$M =\left\{( x , y ) \in R \times R : x ^2+ y ^2 \leq r ^2\right\},$
where $r >0$. Consider the geometric progression $a _{ n }=\frac{1}{2^{ n -1}}, n =1,2,3, \ldots$. Let $S _0=0$ and, for $n \geq 1$, let $S _{ n }$ denote the sum of the first $n$ terms of this progression. For $n \geq 1$, let $C_n$ denote the circle with center $\left(S_{n-1}, 0\right)$ and radius $a _{ n }$, and $D _{ n }$ denote the circle with center $\left( S _{ n -1}, S _{ n -1}\right)$ and radius $a _{ n }$.
($1$) Consider M with $r =\frac{1025}{513}$. Let $k$ be the number of all those circles $C _{ n }$ that are inside $M$. Let $l$ be the maximum possible number of circles among these $k$ circles such that no two circles intersect. Then
$(A)$ $k +2 l=22$ $(B)$ $2 k +l=26$ $(C)$ $2 k +3 l=34$ $(D)$ $3 k +2 l=40$
($2$) Consider $M$ with $r =\frac{\left(2^{199}-1\right) \sqrt{2}}{2^{158}}$. The number of all those circles $D _{ a }$ that are inside $M$ is
$(A) 198$ $(B) 199$ $(C) 200$ $(D) 201$
Give the answer or qution ($1$) and ($2$)