<s>[INST] Choose an infinite GP from given terms for a particular sum Is it possible to choose an infinite GP from amongst the terms 1, 1/2, 1/4, 1/8, 1/16 ... with a sum a) 1/5? b) 1/7 ? 
My approach was simply choosing terms and probable ratios to match the sum...I would be grateful if you could reveal any trick for this question?
 [/INST] Since all terms are of the form $\frac{1}{2^n}$, any geometric progression will be with ratio $\frac{1}{2^k}$
The summation of a geometric series is:
$$\sum\limits_{k=0}^\infty ar^k = \frac{a}{1-r}$$
The question then is what choice of $a$ (initial term) and what choice of $r$ (ratio) will give the desired results.
Since $a$ and $r$ will both be of the form $\frac{1}{2^n}$ we have the sum must be of the form
$$\frac{\frac{1}{2^m}}{1-\frac{1}{2^n}} =\frac{2^{n-m}}{2^n-1}$$
for some given $n$ and $m$
The question is, can you find non-negative integers $n$ and $m$ such that $\frac{2^{n-m}}{2^n-1}$ is equal to $\frac{1}{5}$?  is equal to $\frac{1}{7}$?  If not, then why?  If yes, which?

For $\frac{1}{5}$, the question becomes whether we can find $a,b$ such that $5\cdot 2^a = 2^b-1$.  Notice that for $a\geq 1$ and $b\geq 1$ the left side is even while the right side is odd.  If $b=0$ the right side is zero and the left side is positive.  If $a=0$, the left side is equal to $5$, but no integer $b$ satisfies $2^b-1=5$
Thus $\frac{1}{5}$ is impossible.
For $\frac{1}{7}$, we ask if we can find $a,b$ such that $7\cdot 2^a=2^b-1$.  From the same logic, we see that they cannot both be positive and $b$ cannot be zero.  This leaves $a=0$ and we find that $b=3$ works.
This corresponds back to our $n$ and $m$ from before implying the geometric progression $\frac{1}{8},\frac{1}{64},\frac{1}{512},\dots$ has sum equal to $\frac{1}{7}$
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