To find f(n+1) - f(n-1) in-terms of f(n)

Expert replies

gmattesttaker2: Legendary Member; Posts: 641; Joined: Tue Feb 14, 2012 3:52 pmFollowed by: 8 members; Thanked: 11 times

by gmattesttaker2 » Sun May 04, 2014 9:50 pm

Hello,

Can you please tell me how to solve this:

If n is a positive integer greater than 2 and f(n) = (1+√5)^n/2^n, what is f(n+1)−f(n−1) in terms of f(n)?

A) 1/2 (f(n))

B) sq. root ( f(n) )

C) f(n)

D) 2 f(n)

E) ( f(n) )^2

OA: [spoiler]f(n)[/spoiler]

f(n) = (1 + sq. root 5)^n/2^n = ( (1 + sq. root 5)/2 )^n

f(n + 1) = ( (1 + sq. root 5)/2 )^(n + 1)
f(n - 1) = ( (1 + sq. root 5)/2 )^(n - 1)

f(n + 1) - f(n - 1) = ( (1 + sq. root 5)/2 )^(n + 1) - ( (1 + sq. root 5)/2 )^(n - 1)

I was not able to solve from here onwards. The official explanation mentions about some exponent rule of factoring out the smaller power but I am not very familiar with that. Can you please assist? Thanks a lot for your help.

Best Regards,
Sri

Source: Beat The GMAT — Problem Solving | Sun May 04, 2014 9:50 pm

GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Mon May 05, 2014 3:06 am

If n is a positive integer greater than 2 and f(n) = (1+√5)^n/2^n, what is f(n+1)-f(n-1) in terms of f(n)?

A) 1/2 (f(n))

B) sq. root ( f(n) )

C) f(n)

D) 2 f(n)

E) ( f(n) )^2

The constraint that n>2 is irrelevant, since there is no value of n that will yield a nonreal value for f(n) = (1+√5)^n/2^n.
Thus, we can plug in ANY VALUE for n.

Let n=1.
Then:
f(n+1) = f(2) = (1+√5)²/2² = (1 + 2√5 + 5)/4 = (6 + 2√5)/4 = (3 + √5)/2.
f(n-1) = f(0) = (1+√5)�/2� = 1/1 = 1.
Thus:
f(n+1) - f(n-1) = [ (3 + √5)/2 ] - 1 = [ (3 + √5)/2 ] - (2/2) = (1 + √5)/2. This is our target.

Now plug n=1 into the answers to see which yields our target of (1 + √5)/2.
Since all of the answer choices are in terms of f(n) -- the value in answer choice C -- start with C:
f(n) = f(1) = (1+√5)¹/2¹ = (1 + √5)/2.
Success!

The correct answer is C.

Last edited by GMATGuruNY on Mon May 05, 2014 10:13 am, edited 2 times in total.

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

feedrom: Junior | Next Rank: 30 Posts; Posts: 27; Joined: Fri Nov 15, 2013 10:19 am; Thanked: 4 times

by feedrom » Mon May 05, 2014 9:16 am

Hi Mitch,

The question said that: If n is a positive integer greater than 2, so it's necessary to n>2?

Thank you,

GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Mon May 05, 2014 10:11 am

feedrom wrote:Hi Mitch,

The question said that: If n is a positive integer greater than 2, so it's necessary to n>2?

Thank you,

I've amended my post above to include the following:

The constraint that n>2 is irrelevant, since there is no value of n that will yield a nonreal value for f(n) = (1+√5)^n/2^n.
Thus, we can plug in ANY VALUE for n.

Featured schoolRound 2 closes Jan 7, 2027 53 days left

Chicago Booth

Since 1982, 10 Booth faculty members have received the Nobel Prize in Economic Sciences, more than any other business school.

#3
U.S. News rank: $175,000
Median base salary: 670
Average GMAT Focus: 635
Students in the class

Apply to Booth Visit chicagobooth.edu See the full profile

Photo: Warren LeMay, CC BY-SA 2.0

GMAT Course Reviews

563 reviews

1168 reviews

431 reviews

359 reviews

147 reviews

5/5

5 Star (563 Reviews)

"Target Test Prep is the closest to the official version of the GMAT exam, about 99% accuracy in terms of the quality and quantity of information. The course has excellently created singular sets of focused lessons and tests for every possible topic that one could come across in the official GMAT exam."

"The TTP course maximizes the efficiency of the time you spend studying. It will take time and effort but I could almost guarantee that if you complete the course exactly as it is laid out you will get an amazing score. They also have a very responsive team willing to help with any questions you might have."

"TTP has two things that I think no other test prep company offers: A teaching approach that reinforces understanding and an attitude that will give you the mental preparedness needed to succeed on the test. TTP gives you a deep understanding of the concept you need to know while teaching you how to think."

Write a Review