Target Test Prep 20% Off Flash Sale is on! Code: FLASH20

Redeem
Target Test Prep
5-Day Free Trial
5-day free, full-access trial TTP

Available with Beat the GMAT members only code

MORE DETAILS
Target Test Prep
Magoosh
Magoosh
Study with Magoosh GMAT prep

Available with Beat the GMAT members only code

MORE DETAILS
Magoosh

In the sequence {A_n}, A1=1 and An = 3An-1 + 1 for all posit

This topic has expert replies
Post new topic Post Reply
Previous Topic Next Topic
User avatar
Elite Legendary Member
Posts: 3991
Joined: Fri Jul 24, 2015 2:28 am
Location: Las Vegas, USA
Thanked: 19 times
Followed by:37 members

In the sequence {A_n}, A1=1 and An = 3An-1 + 1 for all posit

by Max@Math Revolution » Fri Jun 01, 2018 1:24 am

Timer

00:00

Your Answer

A

B

C

D

E

Global Stats

[GMAT math practice question]

In the sequence {A_n}, A1=1 and An = 3An-1 + 1 for all positive integers n. What is the units digit of A100?

A. 0
B. 1
C. 2
D. 3
E. 4
Top
User avatar
GMAT Instructor
Posts: 15539
Joined: Tue May 25, 2010 12:04 pm
Location: New York, NY
Thanked: 13060 times
Followed by:1906 members
GMAT Score:790

by GMATGuruNY » Fri Jun 01, 2018 4:04 am
Max@Math Revolution wrote:[GMAT math practice question]

In the sequence {A_n}, A1=1 and An = 3An-1 + 1 for all positive integers n. What is the units digit of A100?

A. 0
B. 1
C. 2
D. 3
E. 4
A� = 1.
A₂ = 3A� + 1 = (3*1) + 1 = 4.
A₃ = 3A₂ + 1 = (3*4) + 1 = 13.
A₄ = 3A₃ + 1 = (3*13) + 1 = 40.
Aâ‚… = 3Aâ‚„ + 1 = (3*40) + 1 = 121.
A₆ = 3A₅ + 1 = (3*121) + 1 = 364.

The results above indicate that the units digits repeat in a CYCLE OF 4:
1, 4, 3, 0...1, 4, 3, 0...
Implication:
Every 4th term -- A₄, A₈, A�₂...A₉₂, A₉₆, A�₀₀ -- will have a units digit of 0.
Thus, the units digit for A�₀₀ is 0.

The correct answer is A.
Private tutor exclusively for the GMAT and GRE, with over 20 years of experience.
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.

As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.

For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
Student Review #2
Student Review #3
Top

GMAT/MBA Expert

User avatar
GMAT Instructor
Posts: 1462
Joined: Thu Apr 09, 2015 9:34 am
Location: New York, NY
Thanked: 39 times
Followed by:22 members

by Jeff@TargetTestPrep » Sun Jun 03, 2018 5:31 pm
Max@Math Revolution wrote:[GMAT math practice question]

In the sequence {A_n}, A1=1 and An = 3An-1 + 1 for all positive integers n. What is the units digit of A100?

A. 0
B. 1
C. 2
D. 3
E. 4
A(1) = 1

A(2) = 3 + 1 = 4

A(3) = 12 + 1 = 13

A(4) = 39 + 1 = 40

A(5) = 120 + 1 = 121

The pattern of the units digits is 1-4-3-0.

So if k is a multiple of 4, then A(k) has a units digit of zero.

Since 100 is a multiple of 4, the units digit of A(100) = 0.

Answer: A

Jeffrey Miller
Head of GMAT Instruction
[email protected]

Image

See why Target Test Prep is rated 5 out of 5 stars on BEAT the GMAT. Read our reviews
Top
User avatar
Elite Legendary Member
Posts: 3991
Joined: Fri Jul 24, 2015 2:28 am
Location: Las Vegas, USA
Thanked: 19 times
Followed by:37 members

by Max@Math Revolution » Mon Jun 04, 2018 11:23 am
=>
The units digit of A_{100} will be the remainder when A_{100} is divided by 10.
Using the formula A_n = 3A_{n - 1} + 1 yields
A2 = 3A1 + 1 = 3*1 + 1 = 3 + 1 = 4 ~ 4
A3 = 3A2 + 1 = 3*4 + 1 = 13 ~ 3
A4 = 3A3 + 1 ~ 3*3 + 1 = 10 ~ 0
A5 = 3A4 + 1 ~ 3*0 + 1 ~ 1
In general, we have:
A1 ~ A5 ~ ... ~ A4k+1 ~ 1
A2 ~ A6 ~ ... ~ A4k+2 ~ 4
A3 ~ A7 ~ ... ~ A4k+3 ~ 3
A4 ~ A8 ~ ... ~ A4k ~ 0
Since the index (100) of A100 is a multiple of 4, the units digit of A100 is 0.
Therefore, the answer is A.
Answer: A
Top
Post new topic Post Reply
• Page 1 of 1