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

Are You Up For the Challenge: 700 Level Questions

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

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

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..

Vigasimrair · Answer

Very elegant solution!
Thank you.

Posted from my mobile device

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

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

The sequence A is defined by the following relationship:

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

The sequence A is defined by the following relationship:

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards