In a certain sequence, a1 = 4 and an = an-1

Question

In a certain sequence,  a_1=4 , and  a_n=(\frac{1}{2})(a_{n−1})  for all  n > 1 . If k = the sum of the first 127 terms of the sequence, which of the following is true?
A)  6 < k < 8 
B)  8 < k < 10 
C)  10 < k < 12 
D)  12 < k < 14 
E)  14 < k < 16

*See if you can answer this question WITHOUT using any formulas for geometric series.

prabsahi · Accepted Answer

Here is my approach to the above problem

Without Geometric Formula

a1=4 
a2= 1/2(an-1)= 1/2*a1= 2
a3= 1/2*a2= 1/2*2 =1
a4= 1/2*a3= 1/2
a5=1/4 =.25
a6=1/8 approax .125
a7=1/16 approax 0.0625
.....
....

denominator further raised to teh power of n will be almost insignificant since we are already adding in 2nd decimal places..

So addining a1 to a6 = 4+2+1+0.5+0.25 + 0.125 = 7.875  (No need to calculate precisely..approax its value less than 8)

So A seems to be a probable option.

With Geometric Progression Formula:

GP formula

Sum of N terms where r is the constant ratio between each term and a is first term is = a(1-r^n)/(1-r)
 If n is very high we can approax Sum = a/(1-r)

Now applying this formula for our case we get SUM= 4/(1-1/2) = 4*2= 8

(Note we have used r tends to infinity in practical it will be some value and so sum will be something slightly less than 8)

Press Kudos if it helps!!

LeoN88 · Answer

IMO A.

a1 = 4
a2 = 2
a3 = 1 will contribute to the integral part of the sum of the series, i.e. 7.

Rest= 1/2 , 1/4, 1/8 will keep adding the decimal values. So IMO 6<k<8.

BrentGMATPrepNow · Answer

If we apply the formula for finding the terms in the sequence, we get: 
 a_1=4 ,  a_2=2 ,  a_3=1 ,  a_4=\frac{1}{2} ,  a_5=\frac{1}{4} ,  a_6=\frac{1}{8} ,.....

Rather than apply any formulas to the question, we can think of the sum this way:

Imagine that I start with a 8-kg brick of gold and, each day, I give you half of my gold. 
So, on day 1, I give you  4  kg (leaving me with 4 kg)
On day 2, I give you  2  kg (leaving me with 2 kg)
On day 3, I give you  1  kg (leaving me with 1 kg)
On day 4, I give you  0.5  kg
On day 5, I give you  0.25  kg
.
.
.
Notice that the total amount of gold I give you after 127 days is the SAME as the sum of the first 127 terms of the sequence. 
Since I started with an 8-kg brick of gold, and since I always retain some of my gold (after giving you half), the total amount I give you CANNOT exceed 8 kg. 
As such, the correct answer is A

Cheers,
Brent

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In a certain sequence, a1 = 4 and an = an-1

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In a certain sequence, a1 = 4 and an = an-1

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