If a1, a2 be integers such that a1 – a2 + a3 – a4 + ........ +(-1)n-1

Question

If  a_1 ,  a_2 , ... are integers such that  a_1 – a_2 + a_3 – a_4 + ........ +(-1)^{n-1}   a_n = n , for  n ≥ 1 . Then what is the value of  a_{51} + a_{52} + ... + a_{1023}  ?

A. -1
B. 0
C. 1
D. 2
E. 10

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chetan2u · Answer

a_1−a_2+a_3−a_4+⋯+(−1)^{(n−1)} a_n=n

Now we have to find a pattern in this. You can easily get the value of  a_1  by substituting n=1..
n=1....... (−1)^{(n−1)} a_n=n....-1^{(1-1)}a_1=1....a_1=1 
n=2....... a_1- a_2=2.....1-a_2=2.....a_2=-1 
n=3....... a_1- a_2+a_3=3.....1-(-1)+a_3=3.....a_3=3-2=1 
n=4....... a_1- a_2+a_3-a_4=4.....1-(-1)+1-a_4=4.....a_4=3-4=-1

So we have a pattern..
 a_{odd}=1 ..
 a_{even}=-1 ..

Now in  a_{51}+a_{52}+⋯+a_{1022}+a_{1023} , we have equal number of even and odd terms  a_{51}+a_{52}+⋯+a_{1022} , so their sum will be 0..
Only term left is  a_{1023} , which is  a_{odd}=1 ..

C

preetamsaha · Answer

a1=1
a1-a2= 2
or, a2=-1
a1-a2+a3= 3
or, a3=1
a1-a2+a3-a4= 4
or, a4=-1
so.  an = (−1)^n−1

a51+a52+⋯+a1022+a1023
=1 (only survives a1023 which is equal to 1)

correct answer C

Kinshook · Answer

Asked: If  a_1 ,  a_2 , ... are integers such that  a_1 – a_2 + a_3 – a_4 + ........ +(-1)^{n-1}   a_n = n , for  n ≥ 1 . Then what is the value of  a_{51} + a_{52} + ... + a_{1023}  ?

a_1 = 1 
 a_1 - a_2 = 2 
 a_2 = a_1 - 2 = 1 - 2 = - 1 
 a_1 - a_2 + a_ 3 = 3 
 a_3 =3-2 = 1
 a_1 - a_2 + a_ 3 - a_4  = 4 
 a_4 = -1 
 a_n = (-1)^n

a_{51} + a_{52} + ... + a_{1023} = 1-1+ 1- 1+ ......+1 = 1

IMO C

ScottTargetTestPrep · Answer

Let’s first determine the first few terms and hope we can discover a pattern for the terms:

a1 = 1,

a1 - a2 = 2 → 1 - a2 = 2 → a2 = -1

a1 - a2 + a3 = 3 → 1 - (-1) + a3 = 3 → a3 = 1

a1 - a2 + a3 - a4 = 4 → 1 - (-1) + 1 - a4 = 4 → a4 = -1

We see that the odd numbered terms are 1 and the even numbered terms are -1. Thus, the sum of a pair of consecutive terms is 0. So we have:

(a51 + a52) + (a53 + a54) + … + (a1021 + a1022) + a1023

0 + 0 + … + 0 + 1

1

Answer: C

Dhwanii · Answer

Can't understand how we got A1, A2, A3 in the first place

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If a1, a2 be integers such that a1 – a2 + a3 – a4 + ........ +(-1)n-1

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If a1, a2 be integers such that a1 – a2 + a3 – a4 + ........ +(-1)n-1

Prep Toolkit

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My Rewards