• Veritas Prep
    Free Veritas GMAT Class
    Experience Lesson 1 Live Free

    Available with Beat the GMAT members only code

    MORE DETAILS
    Veritas Prep
  • Varsity Tutors
    Award-winning private GMAT tutoring
    Register now and save up to $200

    Available with Beat the GMAT members only code

    MORE DETAILS
    Varsity Tutors
  • PrepScholar GMAT
    5 Day FREE Trial
    Study Smarter, Not Harder

    Available with Beat the GMAT members only code

    MORE DETAILS
    PrepScholar GMAT
  • Magoosh
    Magoosh
    Study with Magoosh GMAT prep

    Available with Beat the GMAT members only code

    MORE DETAILS
    Magoosh
  • Kaplan Test Prep
    Free Practice Test & Review
    How would you score if you took the GMAT

    Available with Beat the GMAT members only code

    MORE DETAILS
    Kaplan Test Prep
  • e-gmat Exclusive Offer
    Get 300+ Practice Questions
    25 Video lessons and 6 Webinars for FREE

    Available with Beat the GMAT members only code

    MORE DETAILS
    e-gmat Exclusive Offer
  • Target Test Prep
    5-Day Free Trial
    5-day free, full-access trial TTP Quant

    Available with Beat the GMAT members only code

    MORE DETAILS
    Target Test Prep
  • Economist Test Prep
    Free Trial & Practice Exam
    BEAT THE GMAT EXCLUSIVE

    Available with Beat the GMAT members only code

    MORE DETAILS
    Economist Test Prep
  • The Princeton Review
    FREE GMAT Exam
    Know how you'd score today for $0

    Available with Beat the GMAT members only code

    MORE DETAILS
    The Princeton Review
  • EMPOWERgmat Slider
    1 Hour Free
    BEAT THE GMAT EXCLUSIVE

    Available with Beat the GMAT members only code

    MORE DETAILS
    EMPOWERgmat Slider

There is a sequence A(n) where n is a positive integer...

This topic has 1 expert reply and 0 member replies

There is a sequence A(n) where n is a positive integer...

Post Fri Feb 23, 2018 7:57 am
There is a sequence A(n) where n is a positive integer such that A(n+1) = 10 + 0.5A(n). Which of the following is closest to A(1,000)?

A. 15
B. 18
C. 20
D. 25
E. 50

The OA is C.

Please, can any expert explain this PS question for me? I can't get the correct answer. I need your help. Thanks.

  • +1 Upvote Post
  • Quote
  • Flag

GMAT/MBA Expert

ErikaPrepScholar Legendary Member
Joined
20 Jul 2017
Posted:
503 messages
Followed by:
9 members
Upvotes:
86
GMAT Score:
770
Post Fri Feb 23, 2018 10:04 am
Let's try building out the first few integers in our sequence to see if we notice a pattern:

A(n+1) = 10 + 0.5A(n)

A(1) = A(0+1) --> Note: We can't solve for this because we need the value for A(0), which doesn't exist.
A(2)= A(1+2) $$=10+\frac{1}{2}A\left(1\right)$$
A(3) = A(2+1) $$=10+\frac{1}{2}A\left(2\right)=10+\frac{1}{2}\left(10+\frac{1}{2}A\left(1\right)\right)=10+5+\frac{1}{4}A\left(1\right)=15+\frac{1}{4}A\left(1\right)$$
A(4) = A(3+1) $$=10+\frac{1}{2}A\left(3\right)=10+\frac{1}{2}\left(15+\frac{1}{4}A\left(1\right)\right)=15+2.5+\frac{1}{8}A\left(1\right)=17.5+\frac{1}{8}A\left(1\right)$$
A(5) = A(4+1) $ $$=10+\frac{1}{2}A\left(4\right)=10+\frac{1}{2}\left(17.5+\frac{1}{8}A\left(1\right)\right)=17.5+1.25+\frac{1}{16}A\left(1\right)=18.75+\frac{1}{16}A\left(1\right)$$

At this point (or maybe a couple numbers back), we can probably recognize a pattern - for each number we increase, we add 10/(2^n-1) and reduce A(1) by 1/2. Knowing this, we can extend our pattern further:


$$A\left(6\right)=18.75\ +\ 0.625\ +\ \frac{1}{32}A\left(1\right)=19.375+\frac{1}{32}A\left(1\right)$$ $$A\left(7\right)=19.375\ +\ 0.3125\ +\ \frac{1}{64}A\left(1\right)=19.6875+\frac{1}{64}A\left(1\right)$$ $$A\left(8\right)=19.6875\ +\ 0.15625\ +\ \frac{1}{128}A\left(1\right)=19.84375+\frac{1}{124}A\left(1\right)$$

Now we should notice that 1) Our fraction of A(1) is becoming very small very fast, and 2) our number is increasing slower and slower as it approaches 20. Knowing this, we can extrapolate that by the time we reach A(1000), A(1) will be such a small fraction that it will be negligible, and our number will still be close to 20. So A(1000) will be closest to 20, or answer choice C.

_________________


Erika John - Content Manager/Lead Instructor
https://gmat.prepscholar.com/gmat/s/

Get tutoring from me or another PrepScholar GMAT expert: https://gmat.prepscholar.com/gmat/s/tutoring/

Learn about our exclusive savings for BTG members (up to 25% off) and our 5 day free trial

Check out our PrepScholar GMAT YouTube channel, and read our expert guides on the PrepScholar GMAT blog

  • +1 Upvote Post
  • Quote
  • Flag

Top First Responders*

1 GMATGuruNY 108 first replies
2 Brent@GMATPrepNow 41 first replies
3 Rich.C@EMPOWERgma... 35 first replies
4 Jay@ManhattanReview 25 first replies
5 Scott@TargetTestPrep 11 first replies
* Only counts replies to topics started in last 30 days
See More Top Beat The GMAT Members

Most Active Experts

1 image description GMATGuruNY

The Princeton Review Teacher

154 posts
2 image description Brent@GMATPrepNow

GMAT Prep Now Teacher

129 posts
3 image description Scott@TargetTestPrep

Target Test Prep

103 posts
4 image description Rich.C@EMPOWERgma...

EMPOWERgmat

81 posts
5 image description Max@Math Revolution

Math Revolution

80 posts
See More Top Beat The GMAT Experts