The sequence A is defined by the following relationship:

Question

The sequence A is defined by the following relationship:  A_n=A_{n-1}+(-1)^{n+1}(n)^2  for all integer values  n >1 . If  A_1=1  , what is  A_{15}-A_{13}  ?

A. 14
B. 29
C. 169
D. 196
E. 421

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Har8 · Accepted Answer

You should start from the ending, using the sequence formula.

A15 - A13 = A14 + 15^2 - A13 = A13 - 14^2 + 15^2 - A13 = (15-14) * (15+14) = 29

Archit3110 · Answer

for the given function  A_n=A_{n-1}+(-1)^{n+1}(n)^2  
A1=1
A2=-3
A3= 6
A4=-10
A5=15
A6=-21
.
.
.
.
A13= 91 and A15= 120
A15-A13 = 29
IMO B

if there is any other short cut please share..

dips1122 · Answer

A1=1
A2=-3
A3=6
A4=-10

Once we have found out till A4 we can see the pattern for the next term: add n to the absolute value of the previous and reverse the sign. For e.g. A5 = |A4| + 5 (with reversed sign).
So we can then quickly find out:

A5=15
A6=-21
A7=28
A8=-36
.
.
.
A13=91
A14=-105
A15=120

Hence A15-A13 = 29

B

Vigasimrair · Answer

Very elegant solution!
Thank you.

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Apo_orv · Answer

A1 = 1
A2 = -3
A3 = 6
A4= -10
A5=15
once we figure out a pattern it will simple and we Even Dont need to find out what A15 , A14 AND A13 is.

Difference between each term is mod n or |N| i.e. ||A2|-|A1|| =2 , ||A3|-|A2|| = 3, ||A4|-|A3|| =4
A15-A13=||A15|-|A14||+||A14|-|A13|| === 15+14 =29

So Answer is B, and we even didnt need to calculate all the numbers

shubhamjha22 · Answer

An= An-1 + (-1)^(n+1)(n^2)

An-An-1= (-1)^(n+1)(n^2)

A15-A14= (-1) ^(16) (15^2).........Eq a.

A14-A13= (-1)^(15)(14^2)......... Eq b.

Adding Eq a+b

A15-A14+A14-A13= A15-A13= (-1)^16*(15^2)+ (-1^15)(14^2)

15^2-14^2

29.

Mais123 · Answer

A15=A14+15^2 (1) 
A14=A13-14^2 (2)
Just put (2) into (1) 
(1)=>(2) A15=A13-14^2+15^2=> A15-A13=15^2-14^2=(15-14)(15+14)=29

CEdward · Answer

A15 - A13 = A14 + 15^2 - A13 =  A13 - 14^2 + 15^2 - A13 = (15-14) * (15+14) = 29

Can someone explain the bolded portion? I don't follow.

KarishmaB · Answer

Try to plug in some simple values to understand the pattern first.

A_2 = 1 - 2^2 
 A_3 = A2 + 3^2 = 1 - 2^2 + 3^2 
Now we can already see the pattern:

A_4 = 1 - 2^2 + 3^2 - 4^2 
 A_5 = 1 - 2^2 + 3^2 - 4^2 + 5^2 
and so on...

A_{14} = A_{13} - 14^2 
 A_{15} = A_{13} - 14^2 + 15^2

A_{15} - A_{13} = - 14^2 + 15^2 = 29

Answer (B)

MBAHOUSE · Answer

Sequences are generally medium and hard questions at the real GMAT.
This is a wonderful example about this topic.
Generally, the solutions come in a arithmetic way. I did in an algebraic way.
Difference of squares were applied too.

Tiffsootmoot8 · Answer

Once I saw that A1, A2, A3... were in the pattern
1, -3, 6, -10, 15, ...
apart from the alternating signs , An = sum of 1st n natural numbers
for example, A1 = 1, 
A2 = 2+1
A3 = 3+2+1 
All even terms have negative sign while odd terms have positive signs
Hence in this case , A15- A13 should both be positive in sign 
A13 = 13+12+11+...+1
A15 = 15 + 14 + 13 + 12 +...+1
A15 - A13 = 15 + 14 = 29

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The sequence A is defined by the following relationship:

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