If Sn is the sum of the first n terms of a certain sequence

Question

If Sn is the sum of the first n terms of a certain sequence and if Sn = n(n+1) for all positive integers n, what is the third term of the sequence?

A. 3 
B. 4 
C. 6 
D. 8 
E. 9

KarishmaB · Accepted Answer

You are getting confused with the variables.

Usually,

Tn = nth term
T1 = 1st term
T2 = 2nd term
and so on...

Sn = Sum of first n terms of the sequence
Sn = 1st term + 2nd term + 3rd term + ... + nth term
Sn = T1 + T2 + T3 + ....Tn

You are given here that "Sn is the SUM of first n terms..." So you have 
Sn = T1 + T2 + T3 + ....Tn = n(n+1)

So S1 = T1
S2 = T1 + T2
S3 = T1 + T2 + T3
and so on

S1 = T1 = 1*(1+1) = 2
S2 = T1 + T2 = 2 + T2 = 2*(2+1) = 6
So T2 = 4

S3 = T1 + T2 + T3 = 2 + 4 + T3 = 3*(3+1) = 12 
So T3 = 6 
(This is what we wanted)

The third term is 6.

NoHalfMeasures · Answer

S3 = 3*4 = 12
S2 = 2*3 = 6

So the third element of seq = 12 - 6 = 6
Ans : C

PareshGmat · Answer

Let there be 3 terms in the sequence, namely a, b & c

Given that a + b + c = 3 * 4 = 12 .............. (1)

Now say there are 2 terms, namely a & b

So, a + b = 2 * 3 = 6 .......... (2)

Substitute value of a+b from eqn (2) in eqn (1)

6 + c = 12

c = 6

Answer = C

AdmitJA · Answer

Hi All, I'm totally lost and unable to understand this question. :oops:

a) PareshGmat : where is this given in the question: Given that a + b + c = 3 * 4 = 12 .............. (1) b) MensaNumber : how did you get to this: S3 = 3*4 = 12 S2 = 2*3 = 6 So the third element of seq = 12 - 6 = 6 c) What does the third term mean? d) How have we assumed the first number in the sequence to be 1? Is it because that equation holds true for all positive numbers? But the series could start with say a 101! Any help would be greatly appreciated! :-D

vijayk22 · Answer

Approach: Back Solve

Steps:
1. start with C (6)
2. Is 6 product of subsequent numbers ? - yes 6=2 x3
3 Check other options before confirmation - none of them are products of subsequent number

Answer:C

Bunuel · Answer

We are told that the sum of a certain sequence is given by the formula Sn = n(n+1). So, the sum of the first 2 terms is S2 = 2(2+1) = 6, the sum of the first 3 terms is S3 = 3(3 + 1) = 12 and so on. Basically to get the sum of the first n terms we simply should substitute the value of n in the formula.

Does this make sense?

angelfire213 · Answer

i still don't get it. Are we talking 2 different "S"s here. It seams we are saying SN is the sum oft the first n terms and SN is also equal to n(n+1).

If we say S3 is 6 as the answer says, then n is 3 correct? SO S3 should be 3(3+1) = 12.

That's why i was confused on the question as to how to differentiate between the S's and N's.

any help please

AdmitJA · Answer

Nope. We are telling a) SN is the sum oft the first n terms and b) SN has the equation n(n+1) Hope this helps!

manpreetsingh86 · Answer

Slightly different approach.
Sn=n(n+1)
now, suppose if the sequence has 1 term only. then sum will be equal to that term. so S1=first term
put n=1 in the expression we have
S1=1(1+1) =2
thus first term of the sequence =2
now S2=2(2+1)=6
we know that first term of the sequence =2, thus second term will be 6-2 =4
now, S3=3(3+1) =12, 
thus third term will be 12-first term - second term =12-2-4 =6

angelfire213 · Answer

I guess my question is how is S2 different from the 2nd term conceptually. I know to those who get it, this might seem obvious but I don't understand how S2 is 6 and the 2nd term is 4.

ie isn't S2 = the 2nd term?

(I know once the light bulb clicks i'll be like "ohhhhhhh" but help me understand please)

vandyke1234 · Answer

let A 1 , A 2 , A 3  be the first 3 terms
Sum of first 2 terms = A 1 +A 2  = 2(2+1) = 6 ..............(i)
Sum of the three terms = A 1 +A 2 +A 3  = 3(3+1) = 12 ......(ii)

(ii)-(i) = A 3 = 6

ScottTargetTestPrep · Answer

We see that S3 = 3(4) = 12 and S2 = 2(3) = 6. Since S3 is the sum of the first three terms and S2 is the sum of the first two terms, the difference, S3 - S2 = 12 - 6 = 6, must be the value of the third term of the sequence.

Answer: C

Probus · Answer

Hi,
There is a 15 Sec approach for this question

Well this is worth remembering that

Sum of first 'n' odd natural numbers = n^{2} 
sum of first 'n' even natural numbers= n(n+1)or =  n^{2} +n

So question is asking us for what is the third 3 even natural numbers ( 2,4, 6)

6 is the third even natural number .

Hope this helps

Evo23 · Answer

sum of first 'n' even natural numbers= n(n+1)

So question is asking us for what is the third 3 even natural numbers ( 2,4, 6)

chetan2u  
My question is
Zero is an even integer, so shouldn't the first three terms be 0,2,4 ?

chetan2u · Answer

Natural numbers start from 1, so 0 will not be counted. 
Whole number start from 0.

However I doubt GMAT will deal with natural numbers, it will call it positive integers

Deepam10 · Answer

Could anyone please clarify if S(n) - S(n-1) is always equal to a(n)? 
Sum(nth term) - Sum(nth term -1) = nth term

Example:
S4-S3=a3
Similarly, S2-S1=a2
 Bunuel

Bunuel · Answer

Yes, S(n) - S(n-1) = a(n). This is because S(n) = S(n - 1) + a(n), so the difference S(n) - S(n-1) is obviously a(n). Here, S(n-1) is the sum of the first n-1 terms, so, when you add a(n) to tat you get the sum of n terms, which is S(n).

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If Sn is the sum of the first n terms of a certain sequence

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