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kth term - sequence

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by nikhilkatira » Mon Jun 28, 2010 10:07 pm
For every integer k from 1 to 10,inclusive,the kth term of a certain sequence is given by (-1)^k+1 * (1/2^k).If T is the sum of the 1st 10 terms in the sequence then T is,
a)Greater than 2
b)between 1 and 2
c)between ½ and 1
d)between ¼ and ½
e)less than ¼


How to guesstimate answer in the question ?
Best,
Nikhil H. Katira
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by sreak1089 » Mon Jun 28, 2010 10:53 pm
You know that +ve terms are: 1/2, 1/8, 1/32, 1/128, 1/512
-ve terms are: -1/4, -1/16, -1/64, -1/512, -1/1024
Thus the sum of the series: 1/4 + (1/16 + 1/64 + 1/512 + 1/1024)

Sum in the bracket should be less than 1/4, hence the answer should be
b/w 1/4 & 1/2. Hence D
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by amising6 » Mon Jun 28, 2010 10:56 pm
nikhilkatira wrote:For every integer k from 1 to 10,inclusive,the kth term of a certain sequence is given by (-1)^k+1 * (1/2^k).If T is the sum of the 1st 10 terms in the sequence then T is,
a)Greater than 2
b)between 1 and 2
c)between ½ and 1
d)between ¼ and ½
e)less than ¼


How to guesstimate answer in the question ?
now the sequence would be (-1)^k+1 * (1/2^k).
let put k=1 we get
(-1)^1+1 * (1/2^1).
-1+1/2=-1/2
now next term k=2
(-1)^2 +1 * (1/2^2).
1+1/4=5/4
so term 1 +term2=-1/2+5/4=3/4
similiarly k=3 and k=4
will result in positive result as every odd term will be negative which would be lessthan its next immediate term as every even value of k will result in positive
k3+k4
1+1/16=17/16
now adding k1+k2+k3+k4=3/4+17/16=29/16
now onwards every term will contribute something positive so u can conclude
now you can exclude option as c,d,e and b as if you take k1+k2 the value contributed by them was 3/4 which is less than value contributed by k3+k4
so k5 and k6 will contribute value more than k3 and k4
hence b is also out so option A
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by kvcpk » Mon Jun 28, 2010 10:56 pm
nikhilkatira wrote:For every integer k from 1 to 10,inclusive,the kth term of a certain sequence is given by (-1)^k+1 * (1/2^k).If T is the sum of the 1st 10 terms in the sequence then T is,
a)Greater than 2
b)between 1 and 2
c)between ½ and 1
d)between ¼ and ½
e)less than ¼


How to guesstimate answer in the question ?
Is the OA C?
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by nikhilkatira » Mon Jun 28, 2010 11:06 pm
amising6 wrote:
nikhilkatira wrote:For every integer k from 1 to 10,inclusive,the kth term of a certain sequence is given by (-1)^k+1 * (1/2^k).If T is the sum of the 1st 10 terms in the sequence then T is,
a)Greater than 2
b)between 1 and 2
c)between ½ and 1
d)between ¼ and ½
e)less than ¼


How to guesstimate answer in the question ?
now the sequence would be (-1)^k+1 * (1/2^k).
let put k=1 we get
(-1)^1+1 * (1/2^1).
-1+1/2=-1/2
now next term k=2
(-1)^2 +1 * (1/2^2).
1+1/4=5/4
so term 1 +term2=-1/2+5/4=3/4
similiarly k=3 and k=4
will result in positive result as every odd term will be negative which would be lessthan its next immediate term as every even value of k will result in positive
k3+k4
1+1/16=17/16
now adding k1+k2+k3+k4=3/4+17/16=29/16
now onwards every term will contribute something positive so u can conclude
now you can exclude option as c,d,e and b as if you take k1+k2 the value contributed by them was 3/4 which is less than value contributed by k3+k4
so k5 and k6 will contribute value more than k3 and k4
hence b is also out so option A
OA is D
Best,
Nikhil H. Katira
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by sreak1089 » Mon Jun 28, 2010 11:12 pm
@Nikhil, did u follow my solution?
I am not sure if there is even faster method..
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by amising6 » Mon Jun 28, 2010 11:19 pm
nikhilkatira wrote:
amising6 wrote:
nikhilkatira wrote:For every integer k from 1 to 10,inclusive,the kth term of a certain sequence is given by (-1)^k+1 * (1/2^k).If T is the sum of the 1st 10 terms in the sequence then T is,
a)Greater than 2
b)between 1 and 2
c)between ½ and 1
d)between ¼ and ½
e)less than ¼


How to guesstimate answer in the question ?
now the sequence would be (-1)^k+1 * (1/2^k).
let put k=1 we get
(-1)^1+1 * (1/2^1).
-1+1/2=-1/2
now next term k=2
(-1)^2 +1 * (1/2^2).
1+1/4=5/4
so term 1 +term2=-1/2+5/4=3/4
similiarly k=3 and k=4
will result in positive result as every odd term will be negative which would be lessthan its next immediate term as every even value of k will result in positive
k3+k4
1+1/16=17/16
now adding k1+k2+k3+k4=3/4+17/16=29/16
now onwards every term will contribute something positive so u can conclude
now you can exclude option as c,d,e and b as if you take k1+k2 the value contributed by them was 3/4 which is less than value contributed by k3+k4
so k5 and k6 will contribute value more than k3 and k4
hence b is also out so option A
OA is D
ohh i have solved question can you give question properly
i am solving wrong question this is the way to solve
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by nikhilkatira » Tue Jun 29, 2010 12:13 am
sreak1089 wrote:@Nikhil, did u follow my solution?
I am not sure if there is even faster method..
yup..its an educated guess..
Best,
Nikhil H. Katira
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by kvcpk » Tue Jun 29, 2010 12:22 am
Answer should be D. I read the question wrong earlier.

let me try to simplify this question.
(-1)^k+1 * (1/2^k)

(-1)^k+1 = (-1)(-1)^k = (-1)(-1)^-k [because (-1)^n = (-1)^-n]
(1/2^k) = (2^-k)

now, (-1)^k+1 * (1/2^k) = (-1)(-1)^-k * (2^-k)
= (-1)(-2)^-k
So sum is -(-2)^-1 + -(-2)^-2 + -(-2)^-3 +.................. + -(-2)^-10
= 1/2 - (-1/2)^2 + (-1/2)^3 + ..............
=1/2 -1/4+1/8-1/16+..................

This is clearly a geometric progression with a=1/2 qand r = -1/2

So sum of gometric series is a(r^n -1) /r-1
Using this we get sum as
1/3(1-1/2^10)
= 1/3 - 1/(3*2^10)
this value should be definitely less than 1/3
So answer is in between D,E
Now we know that 1/3-1/12 gives 1/4.
So we need to see if the second half is coming any closer to 1/12. Clearly it is very much less than 1/12 because 1/12 = 1/(3*2^2) and we have 1/(3*2^10)

So the value should be between 1/3 and 1/4. Precisely, we do not have 1/3 in the answers. So Answer should be D.
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