M17 Q05

Question

The value of  \frac{1}{2} + \left(\frac{1}{2}\right)^2 + \left(\frac{1}{2}\right)^3 + ... + \left(\frac{1}{2}\right)^{20}  is between?

A.  \frac{1}{2}  and  \frac{2}{3} 
B.  \frac{2}{3}  and  \frac{3}{4} 
C.  \frac{3}{4}  and  \frac{9}{10} 
D.  \frac{9}{10}  and  \frac{10}{9} 
E.  \frac{10}{9}  and  \frac{3}{2}

EvaJager · Answer

Imagine you have to walk from 0 to 1. You walk half of the way, which is 1/2, and 1/2 left.
You walk now 1/2 of what remained, which is 1/4 and 1/4 remained.
Then you walk 1/2 of what remained, which is 1/8 and 1/8 remained.
.....
The best way to understand is by drawing a line segment and visualizing the distances.
Hope you got the idea. After 20 such walks, you are awfully close to 1, but left of it.
Hence, Answer D.

abhishekkpv · Answer

Adding the first two terms

1/2+1/4 =3/4 or .75 or (2^2-1)/2^2

Adding three terms we get

1/2+1/4+1/8 =7/8 or .875 or (2^3-1)/2^3

Adding four terms

1/2+1/4+1/8+1/16 =15/16 >.9 
15/16 is also (2^4-1)/2^4

so adding all the terms we will get (2^20-1)/2^20 and this is <1
and we already know that this is >.9 as the sum of four terms itself is >.9

and the only choice which satisfies this is D

Bunuel · Answer

The value of  \frac{1}{2} + \left(\frac{1}{2}\right)^2 + \left(\frac{1}{2}\right)^3 + ... + \left(\frac{1}{2}\right)^{20}  is between?

A.  \frac{1}{2}  and  \frac{2}{3} 
B.  \frac{2}{3}  and  \frac{3}{4} 
C.  \frac{3}{4}  and  \frac{9}{10} 
D.  \frac{9}{10}  and  \frac{10}{9} 
E.  \frac{10}{9}  and  \frac{3}{2}

We have the sum of a geometric progression with the first term equal to  \frac{1}{2}  and the common ratio also equal to  \frac{1}{2} .

Now, the sum of  infinite geometric progression  with  common ratio  |r|<1  , is  sum=\frac{b}{1-r} , where  b  is the first term. So, if we had infinite geometric progression instead of just 20 terms then its sum would be  Sum=\frac{\frac{1}{2}}{1-\frac{1}{2}}=1 . Which means that the sum of this sequence will never exceed 1, also as we have big enough number of terms (20) then the sum will be very close to 1, so we can safely choose answer choice D.

Answer: D.

One can also use direct formula.
We have geometric progression with  b=\frac{1}{2} ,  r=\frac{1}{2}  and  n=20 ;

S_n=\frac{b(1-r^n)}{(1-r)}  -->  S_{20}=\frac{\frac{1}{2}(1-\frac{1}{2^{20}})}{(1-\frac{1}{2})}=1-\frac{1}{2^{20}} . Since  \frac{1}{2^{20}}  is very small number then  1-\frac{1}{2^{20}}  will be less than 1 but very close to it.

Answer: D.

vinnik · Answer

Thanks everyone. The methods are just amazing.

Bunuel, your direct formula method is fantastic.

Thanks & Regards
Vinni

M17 Q05

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