Last visit was: 05 Sep 2026, 10:08 It is currently 05 Sep 2026, 10:08
Home
Messages
Tests
Schools
Events
Close
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Go to My Error Log Learn more
Close
Request Expert Reply
Confirm Cancel
Back to Forum
Create Topic
505-555 (Easy)|   Sequences|                           
User avatar
udaymathapati
User avatar
Joined: 06 Apr 2010
Last visit: 27 Jan 2015
Posts: 91
Own Kudos:
5,800
 [232]
Given Kudos: 15
Products:
My Reviews
Posts: 91
Kudos: 5,800
 [232]
The infinite sequence a1, a2, …, an, … is such that a1=2 30 Aug 2010, 10:26
28
Kudos
Add Kudos
203
Bookmarks
Bookmark this Post
00:00
Start Timer
Pause Timer
Resume Timer
Show Answer
Hide
Show
History
B
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!

Difficulty:

15% (low)

Question Stats:

86% (02:05) correct 14% (02:30) wrong based on 3455 sessions

History

Date
Time
Result
Not Attempted Yet
The infinite sequence \(a_1\), \(a_2\),..., \(a_n\), ... is such that \(a_1 = 2\), \(a_2 = -3\), \(a_3 = 5\), \(a_4 = -1\), and \(a_n = a_{n-4}\) for n > 4. What is the sum of the first 97 terms of the sequence?

A. 72
B. 74
C. 75
D. 78
E. 80
ShowHide Answer
Official Answer
B
If this question felt shaky, try an adaptive mini quiz of similar problems in GMAT Club Forum Quiz →. Free plan gives 5 questions per day.
Most Helpful Reply
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 05 Sep 2026
Posts: 113,150
Own Kudos:
839,316
 [66]
Given Kudos: 111,330
Products:
GMAT Club Tests
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 113,150
Kudos: 839,316
 [66]
The infinite sequence a1, a2, …, an, … is such that a1=2 30 Aug 2010, 11:19
30
Kudos
Add Kudos
36
Bookmarks
Bookmark this Post
udaymathapati
The infinite sequence a1, a2,…, an,… is such that a1 = 2, a2 = -3, a3 = 5, a4 = -1, and an = an-4 for n > 4. What is the sum of the first 97 terms of the sequence?
A. 72
B. 74
C. 75
D. 78
E. 80
Show more

Given: \(a_1=2\), \(a_2=-3\), \(a_3=5\) and \(a_4=-1\). Also \(a_n=a_{n-4}\), for \(n>4\).

Since \(a_n=a_{n-4}\) then:
\(a_5=a_1=2\);
\(a_6=a_2=-3\);
\(a_7=a_3=5\);
\(a_8=a_4=-1\);
\(a_9=a_5=a_1=2\);
and so on.

So we have groups of 4: {2, -3, 5, -1}, the sum of each of such group is \(2-3+5-1=3\). 97 terms consist of 24 full groups plus \(a_{97}=a_1=2\), so the sum of first 97 terms of the sequence is \(24*3+2=74\).

Answer: B.
User avatar
KarishmaB
User avatar
Joined: 16 Oct 2010
Last visit: 05 Sep 2026
Posts: 16,643
Own Kudos:
81,182
 [37]
Given Kudos: 491
Location: Pune, India
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 16,643
Kudos: 81,182
 [37]
The infinite sequence a1, a2, …, an, … is such that a1=2 28 Nov 2011, 02:54
23
Kudos
Add Kudos
14
Bookmarks
Bookmark this Post
ashiima
The infinite sequence a1, a2,�, an,� is such that a1 = 2, a2 = -3, a3 = 5, a4 = -1, and an =
an-4 for n > 4. What is the sum of the first 97 terms of the sequence?
A. 72
B. 74
C. 75
D. 78
E. 80
Show more

Given: a(n) = a(n-4) i.e. the nth term is same as (n-4)th term e.g. 5th term is same as 1st term. 6th term is same as 2nd term etc

The sequence becomes: 2, -3, 5, -1, 2, -3, 5, -1, 2, -3, 5, -1 ...
The sum of each group of 4 terms = 2-3+5-1 = 3
How many such groups of 4 are there in the first 97 terms? When you divide by 4, you get 24 as quotient and 1 as remainder.
This means you get 24 complete groups of 4 terms each and 1 extra term i.e. the 97th term.
Sum of 24 groups of 4 terms each = 24*3 = 72
The 97th term will be 2 since it is the first term of the next group of four terms.
Sum of first 97 terms = 72+2 = 74
General Discussion
User avatar
ankitranjan
User avatar
Joined: 08 Sep 2010
Last visit: 13 Dec 2012
Posts: 109
Own Kudos:
878
 [15]
Given Kudos: 21
Location: India
Concentration: Finance
WE 1: 6 Year, Telecom(GSM)
Posts: 109
Kudos: 878
 [15]
The infinite sequence a1, a2, …, an, … is such that a1=2 25 Oct 2010, 04:01
11
Kudos
Add Kudos
4
Bookmarks
Bookmark this Post
the sequence is repeating after each 4 terms.

i.e 2,-3,5,-1 ,2,-3,5,-1 ,2,-3,5,-1 ,2,-3,5,-1 ....till 97th term.

Now (2,-3,5,-1 ),(2,-3,5,-1) ,(2,-3,5,-1 ),......................(2,-3,5,-1) ,1

97 = 24 *4 + 1

Means 24 similar like (2,-3,5,-1 ) and one more that is first term of the series

the sum of the first four term is = 3


Hence sum is 24*3 + 2 = 74

Hence answer is B.

Consider KUDOS if it is helpful to u.
User avatar
shinbhu
User avatar
Joined: 29 Jul 2011
Last visit: 14 Oct 2013
Posts: 51
Own Kudos:
178
 [2]
Given Kudos: 6
Location: United States
Posts: 51
Kudos: 178
 [2]
The infinite sequence a1, a2, …, an, … is such that a1=2 04 Jan 2012, 11:26
1
Kudos
Add Kudos
1
Bookmarks
Bookmark this Post
Nice one, a pattern problem with an addition twist.

a1 = 2, a2 = -3, a3 = 5, a4 = -1
What an = a(n-4) means that from a5 onwards, every number in the sequence will be equal to 4 numbers preceding that number. Therefore, a5 = a1 = 2, a6 = a2 = -3,...

Notice that this is a 4-element set pattern. If we sum a1-a4, we get 3. What is the closet multiple of 4 to 97 - 96! So, there are 24 such complete sets. Added to that is 97th number, which is first in the sequence = 2. So, we get (24 x 3) + 2 = 74 -> B
avatar
PareshGmat
avatar
Joined: 27 Dec 2012
Last visit: 10 Jul 2016
Posts: 1,531
Own Kudos:
8,430
 [4]
Given Kudos: 193
Status:The Best Or Nothing
Location: India
Concentration: General Management, Technology
WE:Information Technology (Computer Software)
Posts: 1,531
Kudos: 8,430
 [4]
The infinite sequence a1, a2, …, an, … is such that a1=2 15 Jul 2014, 20:43
4
Kudos
Add Kudos
Bookmarks
Bookmark this Post
\(a_1 = 2\)

\(a_2 = -3\)

\(a_3 = 5\)

\(a_4 = -1\)

\(a_5 = a_{(5-4)} = a_1 = 2\)

\(a_6 = a_{(6-4)} = a_2 = -3\)

\((a_1 + a_2 + a_3 + a_4) = (a_5 + a_6 + a_7 + a_8) = (a_9 + a_{10} .....)\) and so on

Addition upto \(a_{96}\) is repeated 24 times

\((a_1 + a_2 + a_3 + a_4) = 2 - 3 + 5 - 1 = 3\)

\(a_1 + a_2 + a_3 + ........ + a_{96} = 3 * 24 = 72\)

\((a_1+a_2+a_3+ ........ + a_{96}) + a_{97} = 72 + 2 = 74\)

Answer = B
User avatar
ScottTargetTestPrep
User avatar
Target Test Prep Representative
Joined: 14 Oct 2015
Last visit: 05 Sep 2026
Posts: 22,730
Own Kudos:
26,938
 [2]
Given Kudos: 302
Status:Founder & CEO
Affiliations: Target Test Prep
Location: United States (CA)
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 22,730
Kudos: 26,938
 [2]
The infinite sequence a1, a2, …, an, … is such that a1=2 02 Mar 2017, 18:13
2
Kudos
Add Kudos
Bookmarks
Bookmark this Post
udaymathapati
The infinite sequence \(a_1\), \(a_2\),…, \(a_n\), … is such that \(a_1 = 2\), \(a_2 = -3\), \(a_3 = 5\), \(a_4 = -1\), and \(a_n = a_{n-4}\) for n > 4. What is the sum of the first 97 terms of the sequence?

A. 72
B. 74
C. 75
D. 78
E. 80
Show more

We are given that a(1) = 2, a(2) = -3, a(3) = 5, and a(4) = -1.

Since a(n) = a(n-4), we see that:

a(5) = a(5-4) = a(1) = 2,

a(6) = a(6-4) = a(2) = -3,

a(7) = a(7-4) = a(3) = 5,

and a(8) = a(8-4) = a(4) = -1

As we can see, the terms repeat themselves in a cycle of 4.

Since the sum of the terms a(1) to a(4) inclusive is [2 + (-3) + 5 + (-1)] = 3, the sum of a(5) to a(8) will also be 3, as will the sum of a(9) to a(12), etc.

We see from each “grouping” of 4 numbers, we have a sum of 3, and from 1 to 96 inclusive there are:

(96 - 4)/4 + 1 = 92/4 + 1 = 24 such groupings. Thus, the sum of those 24 groupings is 24 x 3 = 72.

We must also add the value of a(97), which equals 2.

Thus, the sum of the first 97 terms is: 24 x 3 + 2 = 72 + 2 = 74.

Answer: B
User avatar
dave13
User avatar
Joined: 09 Mar 2016
Last visit: 10 Jul 2026
Posts: 1,081
Own Kudos:
1,161
Given Kudos: 3,851
Posts: 1,081
Kudos: 1,161
The infinite sequence a1, a2, …, an, … is such that a1=2 28 Feb 2018, 10:14
Kudos
Add Kudos
Bookmarks
Bookmark this Post
udaymathapati
The infinite sequence \(a_1\), \(a_2\),…, \(a_n\), … is such that \(a_1 = 2\), \(a_2 = -3\), \(a_3 = 5\), \(a_4 = -1\), and \(a_n = a_{n-4}\) for n > 4. What is the sum of the first 97 terms of the sequence?

A. 72
B. 74
C. 75
D. 78
E. 80
Show more

Can anyone provide me with a quality information(website, youtube etc) on GMAT sequences. I cant seem to decode such sort of questions :)
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 05 Sep 2026
Posts: 113,150
Own Kudos:
839,316
 [2]
Given Kudos: 111,330
Products:
GMAT Club Tests
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 113,150
Kudos: 839,316
 [2]
The infinite sequence a1, a2, …, an, … is such that a1=2 28 Feb 2018, 10:18
Kudos
Add Kudos
2
Bookmarks
Bookmark this Post
dave13
udaymathapati
The infinite sequence \(a_1\), \(a_2\),…, \(a_n\), … is such that \(a_1 = 2\), \(a_2 = -3\), \(a_3 = 5\), \(a_4 = -1\), and \(a_n = a_{n-4}\) for n > 4. What is the sum of the first 97 terms of the sequence?

A. 72
B. 74
C. 75
D. 78
E. 80

Can anyone provide me with a quality information(website, youtube etc) on GMAT sequences. I cant seem to decode such sort of questions :)
Show more

12. Sequences



For ALL other subjects check Ultimate GMAT Quantitative Megathread

(I pointed you to this topic multiple times)

Hope it helps.
User avatar
Sri07
User avatar
Joined: 27 Jun 2019
Last visit: 30 Aug 2021
Posts: 41
Own Kudos:
9
Given Kudos: 165
Posts: 41
Kudos: 9
The infinite sequence a1, a2, …, an, … is such that a1=2 28 Jun 2020, 23:51
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Bunuel
udaymathapati
The infinite sequence a1, a2,…, an,… is such that a1 = 2, a2 = -3, a3 = 5, a4 = -1, and an = an-4 for n > 4. What is the sum of the first 97 terms of the sequence?
A. 72
B. 74
C. 75
D. 78
E. 80

Given: \(a_1=2\), \(a_2=-3\), \(a_3=5\) and \(a_4=-1\). Also \(a_n=a_{n-4}\), for \(n>4\).

Since \(a_n=a_{n-4}\) then:
\(a_5=a_1=2\);
\(a_6=a_2=-3\);
\(a_7=a_3=5\);
\(a_8=a_4=-1\);
\(a_9=a_5=a_1=2\);
and so on.

So we have groups of 4: {2, -3, 5, -1}, the sum of each of such group is \(2-3+5-1=3\). 97 terms consist of 24 full groups plus \(a_{97}=a_1=2\), so the sum of first 97 terms of the sequence is \(24*3+2=74\).

Answer: B.
Show more


Hi can you help understand why can't we use the formula here N(A1+AN)/2 ? Thanks
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 05 Sep 2026
Posts: 113,150
Own Kudos:
839,316
 [1]
Given Kudos: 111,330
Products:
GMAT Club Tests
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 113,150
Kudos: 839,316
 [1]
The infinite sequence a1, a2, …, an, … is such that a1=2 29 Jun 2020, 00:05
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Sri07
Bunuel
udaymathapati
The infinite sequence a1, a2,…, an,… is such that a1 = 2, a2 = -3, a3 = 5, a4 = -1, and an = an-4 for n > 4. What is the sum of the first 97 terms of the sequence?
A. 72
B. 74
C. 75
D. 78
E. 80

Given: \(a_1=2\), \(a_2=-3\), \(a_3=5\) and \(a_4=-1\). Also \(a_n=a_{n-4}\), for \(n>4\).

Since \(a_n=a_{n-4}\) then:
\(a_5=a_1=2\);
\(a_6=a_2=-3\);
\(a_7=a_3=5\);
\(a_8=a_4=-1\);
\(a_9=a_5=a_1=2\);
and so on.

So we have groups of 4: {2, -3, 5, -1}, the sum of each of such group is \(2-3+5-1=3\). 97 terms consist of 24 full groups plus \(a_{97}=a_1=2\), so the sum of first 97 terms of the sequence is \(24*3+2=74\).

Answer: B.


Hi can you help understand why can't we use the formula here N(A1+AN)/2 ? Thanks
Show more

The formula you quote, is for evenly spaced sets (aka arithmetical progression), and cannot be applied to all sequences. The sequence at hand is 2, -3, , 5, -1, ... As you can see this is not an arithmetic progression, so you cannot use that formula.

Check links below for more.


12. Sequences



For other subjects:
ALL YOU NEED FOR QUANT ! ! !
Ultimate GMAT Quantitative Megathread

Hope it helps.
User avatar
nicolasnemtala
User avatar
Joined: 13 Dec 2021
Last visit: 13 Jan 2023
Posts: 5
Own Kudos:
1
Given Kudos: 74
Posts: 5
Kudos: 1
The infinite sequence a1, a2, …, an, … is such that a1=2 01 Jun 2022, 07:32
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Hello,

So this progression is neither an AP or GP correct? and thus the equations for the sum do not apply
User avatar
wickedvikram
User avatar
Joined: 08 Jun 2008
Last visit: 21 Jul 2026
Posts: 198
Own Kudos:
107
Given Kudos: 101
Location: India
Concentration: Finance, General Management
Schools: INSEAD Jan '26
GPA: 6.5
Products:
My Reviews
Schools: INSEAD Jan '26
Posts: 198
Kudos: 107
The infinite sequence a1, a2, …, an, … is such that a1=2 24 Mar 2026, 23:56
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Technically it is 4 APs with d=0 after every 4 terms.

nicolasnemtala
Hello,

So this progression is neither an AP or GP correct? and thus the equations for the sum do not apply
Show more
User avatar
PGTLrowanhand
User avatar
Joined: 30 Oct 2012
Last visit: 04 Sep 2026
Posts: 161
Own Kudos:
189
Given Kudos: 3
Status:London UK GMAT Consultant / Tutor
Expert
Expert reply
Posts: 161
Kudos: 189
The infinite sequence a1, a2, …, an, … is such that a1=2 16 Jul 2026, 14:39
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Here's my video solution:

udaymathapati
The infinite sequence \(a_1\), \(a_2\),..., \(a_n\), ... is such that \(a_1 = 2\), \(a_2 = -3\), \(a_3 = 5\), \(a_4 = -1\), and \(a_n = a_{n-4}\) for n > 4. What is the sum of the first 97 terms of the sequence?

A. 72
B. 74
C. 75
D. 78
E. 80
Show more
Moderator:
Bunuel
Math Expert
113150 posts