Last visit was: 14 Jul 2024, 21:56 It is currently 14 Jul 2024, 21:56
Toolkit
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

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.

# If S(n) is the sum of sequence 1, 2, 3, 4, ...n, in terms of

SORT BY:
Tags:
Show Tags
Hide Tags
Manager
Joined: 16 Feb 2011
Posts: 142
Own Kudos [?]: 1009 [70]
Given Kudos: 9
Manager
Joined: 04 Jun 2011
Posts: 99
Own Kudos [?]: 164 [10]
Given Kudos: 21
Director
Joined: 22 Mar 2013
Status:Everyone is a leader. Just stop listening to others.
Posts: 606
Own Kudos [?]: 4647 [7]
Given Kudos: 235
Location: India
GPA: 3.51
WE:Information Technology (Computer Software)
General Discussion
Intern
Joined: 02 Aug 2011
Posts: 6
Own Kudos [?]: 3 [0]
Given Kudos: 1
Re: If S(n) is the sum of sequence 1, 2, 3, 4, ...n, in terms of [#permalink]
"for starters u could use the substitution technique where n =2 ==> sn = 3

then s(2n) = s(4) = 10 only D satisfies"

Can you explain how you would get s(n) = 3 if n is 2. disregarding the format of the sequence, if n is 2, the sum of the sequence should be at least 12 (10...+ 2).

its clearly much quicker than doing it mathematically! but I did go the math route, and my only falter compared to your calculation is that I cannot see how you've got rid of the division by 2 in the S(n) calculations.

"Therefore Sn = n(1+ n)/2 or n+n^2 = 2Sn -- (1)"

Shouldn't n(1 + n) / 2 become n + n^2 / 2?
Retired Moderator
Joined: 20 Dec 2010
Posts: 1108
Own Kudos [?]: 4752 [7]
Given Kudos: 376
Re: If S(n) is the sum of sequence 1, 2, 3, 4, ...n, in terms of [#permalink]
6
Kudos
1
Bookmarks
DeeptiM wrote:
If S(n) is the sum of sequence 1, 2, 3, 4, ...n, in terms of n and S(n), S(2n)=?
(A) 2*S(n)
(B) n*S(n)
(C) 2n*S(n)
(D) 2S(n)+n^2
(E) S(n)+2n^2

Pls help with the easiest explanation possible..thnx

Let's see the pattern:

For n=5, the sequence will be {1,2,3,4,5}
$$S(n)=S(5)=1+2+3+4+5$$

2n=2*5=10, the sequence will be {1,2,3,4,5,6,7,8,9,10}
$$S(2n)=S(10)=1+2+3+4+5+6+7+8+9+10=1+2+3+4+5+(1+5)+(2+5)+(3+5)+(4+5)+(5+5)$$
$$(1+2+3+4+5)+(1+2+3+4+5)+(5+5+5+5+5)$$
$$S(5)+S(5)+5*5=S(5)+S(5)+5^2=2S(5)+5^2$$

Since, n=5
$$2S(5)+5^2=2S(n)+n^2$$

In general terms,
$$S(n)=1+2+3+4,...+n$$
$$S(2n)=1+2+3+4,...+n+(1+n)+(2+n)+(3+n)+(4+n),...+(n+n)$$
$$S(2n)=(1+2+3+4+...+n)+(1+2+3+4+...n)+(n+n+...n-times)$$
$$S(2n)=S(n)+S(n)+n^2$$
$$S(2n)=2S(n)+n^2$$

Ans: "D"
Manager
Joined: 04 Jun 2011
Posts: 99
Own Kudos [?]: 164 [1]
Given Kudos: 21
Re: If S(n) is the sum of sequence 1, 2, 3, 4, ...n, in terms of [#permalink]
1
Kudos
meshell wrote:
"for starters u could use the substitution technique where n =2 ==> sn = 3

then s(2n) = s(4) = 10 only D satisfies"

Can you explain how you would get s(n) = 3 if n is 2. disregarding the format of the sequence, if n is 2, the sum of the sequence should be at least 12 (10...+ 2).

its clearly much quicker than doing it mathematically! but I did go the math route, and my only falter compared to your calculation is that I cannot see how you've got rid of the division by 2 in the S(n) calculations.

"Therefore Sn = n(1+ n)/2 or n+n^2 = 2Sn -- (1)"

Shouldn't n(1 + n) / 2 become n + n^2 / 2?

Michelle, the series is 1,2,3,4,....
and Sn is the sum of the series until n terms .. so the sum of the series for 2 terms or s(2) = 1+2 = 3

and s(4) = 1+2+3+4 = 10

i hope this helps explain your concern on "disregarding the format of the sequence, if n is 2, the sum of the sequence should be at least 12 (10...+ 2). "
if you still have questions, i'll be happy to help.

on the mathematical formula yes sn = [n(1+n)] / 2 and is therefore indeed sn = [n + n^2] / 2
but to avoid confusion, i have pulled the 2 to the other side making it 2* Sn = [n + n^2]

so (n + n^2 ) equals 2*Sn and not just Sn.
Manager
Joined: 04 Jun 2011
Posts: 99
Own Kudos [?]: 164 [1]
Given Kudos: 21
Re: If S(n) is the sum of sequence 1, 2, 3, 4, ...n, in terms of [#permalink]
1
Bookmarks
fluke wrote:
DeeptiM wrote:
If S(n) is the sum of sequence 1, 2, 3, 4, ...n, in terms of n and S(n), S(2n)=?
(A) 2*S(n)
(B) n*S(n)
(C) 2n*S(n)
(D) 2S(n)+n^2
(E) S(n)+2n^2

Pls help with the easiest explanation possible..thnx

Let's see the pattern:

For n=5, the sequence will be {1,2,3,4,5}
$$S(n)=S(5)=1+2+3+4+5$$

2n=2*5=10, the sequence will be {1,2,3,4,5,6,7,8,9,10}
$$S(2n)=S(10)=1+2+3+4+5+6+7+8+9+10=1+2+3+4+5+(1+5)+(2+5)+(3+5)+(4+5)+(5+5)$$
$$(1+2+3+4+5)+(1+2+3+4+5)+(5+5+5+5+5)$$
$$S(5)+S(5)+5*5=S(5)+S(5)+5^2=2S(5)+5^2$$

Since, n=5
$$2S(5)+5^2=2S(n)+n^2$$

In general terms,
$$S(n)=1+2+3+4,...+n$$
$$S(2n)=1+2+3+4,...+n+(1+n)+(2+n)+(3+n)+(4+n),...+(n+n)$$
$$S(2n)=(1+2+3+4+...+n)+(1+2+3+4+...n)+(n+n+...n-times)$$
$$S(2n)=S(n)+S(n)+n^2$$
$$S(2n)=2S(n)+n^2$$

Ans: "D"

Thanks Fluke for saving my back on so many occasions kudos to u!!
Intern
Joined: 14 Oct 2016
Posts: 31
Own Kudos [?]: 35 [0]
Given Kudos: 155
Location: India
WE:Sales (Energy and Utilities)
Re: If S(n) is the sum of sequence 1, 2, 3, 4, ...n, in terms of [#permalink]
S(n)=n(n+1)/2

S(n)=(n/2 )(n+1)

S(2n)=(2n/2)(2n+1)

S(2n)=n(2n+1)
S(2n)=2n^2+n

S(2n)=(n^2)+n +(n^2)
S(2n)= 2S(n)+ n^2
Manager
Joined: 30 Jul 2014
Status:MBA Completed
Affiliations: IIM
Posts: 91
Own Kudos [?]: 97 [0]
Given Kudos: 107
Re: If S(n) is the sum of sequence 1, 2, 3, 4, ...n, in terms of [#permalink]
I calculated 2*S(n) in place of S(2*n) - silly mistake, and hence landed up in the answer option A.
Manager
Joined: 10 Apr 2018
Posts: 185
Own Kudos [?]: 456 [4]
Given Kudos: 115
Location: United States (NC)
Re: If S(n) is the sum of sequence 1, 2, 3, 4, ...n, in terms of [#permalink]
3
Kudos
Hi,
If S(n) = n(n+1)/ 2
or n^2+n= 2S(n) ....... (i)
then S(2n)= 2n(2n+1)/2
= n(2n+1)
= 2n^2 +n
=n^2+n^2+n
= n^2 + 2S(n) { substituting the value from eq i)

Bunuel,
Can we move this to PS forum.
Intern
Joined: 06 Nov 2016
Posts: 49
Own Kudos [?]: 100 [5]
Given Kudos: 151
Location: Viet Nam
GPA: 3.54
Re: If S(n) is the sum of sequence 1, 2, 3, 4, ...n, in terms of [#permalink]
3
Kudos
2
Bookmarks
DeeptiM wrote:
If S(n) is the sum of sequence 1, 2, 3, 4, ...n, in terms of n and S(n), S(2n)=?
(A) 2*S(n)
(B) n*S(n)
(C) 2n*S(n)
(D) 2S(n)+n^2
(E) S(n)+2n^2

Pls help with the easiest explanation possible..thnx

1. Number plugging approach: Let n = 3, we have
S(n) = S(3) = 1+2+3 = 6
S(2n) = S(6) = 1+2+3+4+5+6 = 21

2. Mathematical approach:
$$S(n) = 1+2+3+...+(n-1)+n = \frac{(n+1)*n}{2}$$
--> $$(n+1)*n = 2* S(n)$$
$$S(2n) = 1+2+3+...+(2n-1)+2n = \frac{(2n+1)*2n}{2}$$ = $$(2n+1)*n$$ = $$(n+1)*n$$ + $$n*n$$ = $$2*S(n)$$ + $$n^2$$

To moderators
This is a PS question. Please move it to PS subforum.
Math Expert
Joined: 02 Sep 2009
Posts: 94342
Own Kudos [?]: 640860 [0]
Given Kudos: 85011
Re: If S(n) is the sum of sequence 1, 2, 3, 4, ...n, in terms of [#permalink]
Probus wrote:
Hi,
If S(n) = n(n+1)/ 2
or n^2+n= 2S(n) ....... (i)
then S(2n)= 2n(2n+1)/2
= n(2n+1)
= 2n^2 +n
=n^2+n^2+n
= n^2 + 2S(n) { substituting the value from eq i)

Bunuel,
Can we move this to PS forum.

_____________
Done. Thank you.
Senior Manager
Joined: 14 Dec 2017
Posts: 419
Own Kudos [?]: 463 [0]
Given Kudos: 173
Location: India
Re: If S(n) is the sum of sequence 1, 2, 3, 4, ...n, in terms of [#permalink]
DeeptiM wrote:
If S(n) is the sum of sequence 1, 2, 3, 4, ...n, in terms of n and S(n), S(2n)=?
(A) 2*S(n)
(B) n*S(n)
(C) 2n*S(n)
(D) 2S(n)+n^2
(E) S(n)+2n^2

Pls help with the easiest explanation possible..thnx

Given $$S(n) = 1 + 2 + 3 +.....+ n$$

$$S(2n) = 1 + 2 + 3 +......+ n + (n+1) + (n+2) +.......+ (n+n)$$

$$S(2n) - S(n) = (n+1) + (n+2) +.......+ (n+n) = n*n + (1 + 2 + 3 +....n)$$

hence $$S(2n) = n^2 + 2S(n)$$

Thanks,
GyM
Director
Joined: 09 Mar 2018
Posts: 776
Own Kudos [?]: 458 [2]
Given Kudos: 123
Location: India
Re: If S(n) is the sum of sequence 1, 2, 3, 4, ...n, in terms of [#permalink]
2
Kudos
DeeptiM wrote:
If S(n) is the sum of sequence 1, 2, 3, 4, ...n, in terms of n and S(n), S(2n)=?
(A) 2*S(n)
(B) n*S(n)
(C) 2n*S(n)
(D) 2S(n)+n^2
(E) S(n)+2n^2

Pls help with the easiest explanation possible..thnx

For this question, i just assumed a series of 5 numbers 1,2,3,4,5
S(n) = 15 and n = 5

S(2n) = 55 ( 1,2,3,4,5,6,7,8,9,10)

Now we need to find 55 when we substitute the values for n and S(n) in the answer options
(A) 2*S(n) => 2* 15 =30
(B) n*S(n) => 5*15 =75
(C) 2n*S(n) => 10*15 =150
(D) 2S(n)+n^2 => 55
(E) S(n)+2n^2

VP
Joined: 11 Aug 2020
Posts: 1247
Own Kudos [?]: 207 [0]
Given Kudos: 332
Re: If S(n) is the sum of sequence 1, 2, 3, 4, ...n, in terms of [#permalink]
fluke wrote:
DeeptiM wrote:
If S(n) is the sum of sequence 1, 2, 3, 4, ...n, in terms of n and S(n), S(2n)=?
(A) 2*S(n)
(B) n*S(n)
(C) 2n*S(n)
(D) 2S(n)+n^2
(E) S(n)+2n^2

Pls help with the easiest explanation possible..thnx

Let's see the pattern:

For n=5, the sequence will be {1,2,3,4,5}
$$S(n)=S(5)=1+2+3+4+5$$

2n=2*5=10, the sequence will be {1,2,3,4,5,6,7,8,9,10}
$$S(2n)=S(10)=1+2+3+4+5+6+7+8+9+10=1+2+3+4+5+(1+5)+(2+5)+(3+5)+(4+5)+(5+5)$$
$$(1+2+3+4+5)+(1+2+3+4+5)+(5+5+5+5+5)$$
$$S(5)+S(5)+5*5=S(5)+S(5)+5^2=2S(5)+5^2$$

Since, n=5
$$2S(5)+5^2=2S(n)+n^2$$

In general terms,
$$S(n)=1+2+3+4,...+n$$
$$S(2n)=1+2+3+4,...+n+(1+n)+(2+n)+(3+n)+(4+n),...+(n+n)$$
$$S(2n)=(1+2+3+4+...+n)+(1+2+3+4+...n)+(n+n+...n-times)$$
$$S(2n)=S(n)+S(n)+n^2$$
$$S(2n)=2S(n)+n^2$$

Ans: "D"

Very tough problem because it takes a lot of time to see the pattern. Are there other questions like this?
Manager
Joined: 23 Aug 2015
Posts: 52
Own Kudos [?]: 29 [0]
Given Kudos: 85
Location: India
GPA: 2.99
Re: If S(n) is the sum of sequence 1, 2, 3, 4, ...n, in terms of [#permalink]
Keep it simple..
Substitute any value for n and calculate S(n) and S(2n)
For example, for n=4, S(4) = 10 and S(8) = 36.
Now try substituting n=4 for each of the options and see which one matches.
Only D works!
Re: If S(n) is the sum of sequence 1, 2, 3, 4, ...n, in terms of [#permalink]
Moderator:
Math Expert
94342 posts