GRE Prep 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
Track
Your Progress
Practice
Pays
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
GRE Prep Club Team Member
Joined: 20 Feb 2017
Posts: 2506
Given Kudos: 1055
GPA: 3.39
The sequence f(n) = (2n)! ÷ n! is defined for all positive integer
[#permalink]
17 Aug 2021, 09:11
17 Aug 2021, 09:11
Expert Reply
5
Bookmarks
00:00
Question Stats:
35% (03:42) correct
64% (03:37) wrong based on 14 sessions
Hide Show timer Statistics
The sequence f(n) = (2n)! ÷ n! is defined for all positive integer values of n. If x is defined as the product of the first 10 ten terms of this sequence, which of the following is the greatest factor of x?
(A) 2^20
(B) 2^30
(C) 2^45
(D) 2^52
(E) 2^55
(A) 2^20
(B) 2^30
(C) 2^45
(D) 2^52
(E) 2^55
Retired Moderator
Joined: 10 Apr 2015
Posts: 6218
Given Kudos: 136
The sequence f(n) = (2n)! ÷ n! is defined for all positive integer
[#permalink]
06 Sep 2021, 08:18
06 Sep 2021, 08:18
2
GeminiHeat wrote:
The sequence f(n) = (2n)! ÷ n! is defined for all positive integer values of n. If x is defined as the product of the first 10 ten terms of this sequence, which of the following is the greatest factor of x?
(A) 2^20
(B) 2^30
(C) 2^45
(D) 2^52
(E) 2^55
(A) 2^20
(B) 2^30
(C) 2^45
(D) 2^52
(E) 2^55
Let's look for a pattern...
Term 1 = 2!/1! = 2 = 2^1
Term 2 = 4!/2! = 12 = 3(2^2)
Term 3 = 6!/3! = 120 = 15(2^3)
Term 4 = 8!/4! = 1680 = 105(2^4)
Continuing we get...
Term 9 = 18!/9! = ???(2^9)
Term 10 = 20!/1! = ???(2^10)
So, if we focus on the 2's, we get: x = (2^1)(2^2)(2^3)(2^4). . . (2^9)(2^10)
= 2^(1+2+3+4+5+6+7+8+9+10)
= 2^(55)
Answer: E
General Discussion
Re: The sequence f(n) = (2n)! ÷ n! is defined for all positive integer
[#permalink]
20 Aug 2021, 15:39
20 Aug 2021, 15:39
1
(2n)!/n! = (1*2*3*......*2n)/n! = (1*3*5*7*......*2n-1) * (2*4*6*8......*2n)/n! = (1*3*5*......*2n-1) * (1*2*3......*n)*2^n /n!
For n=1, 2^n has a power of 1. Similarly, for n=10, 2^n will have a power of 10. We have to add all the powers from n=1 to n=10
Answer= 2^(1+2+3+4+.........+10)=2^55
For n=1, 2^n has a power of 1. Similarly, for n=10, 2^n will have a power of 10. We have to add all the powers from n=1 to n=10
Answer= 2^(1+2+3+4+.........+10)=2^55
