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If Sn is the sum of the first n terms of a certain sequence  [#permalink]

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6
34 00:00

Difficulty:   35% (medium)

Question Stats: 68% (01:32) correct 32% (01:45) wrong based on 474 sessions

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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

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Re: If Sn is the sum of the first n terms of a certain sequence  [#permalink]

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8
6
angelfire213 wrote:
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?

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.
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Re: If Sn is the sum of the first n terms of a certain sequence  [#permalink]

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12
1
S3 = 3*4 = 12
S2 = 2*3 = 6

So the third element of seq = 12 - 6 = 6
Ans : C
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Re: If Sn is the sum of the first n terms of a certain sequence  [#permalink]

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5
2
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

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Re: If Sn is the sum of the first n terms of a certain sequence  [#permalink]

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Hi All,

I'm totally lost and unable to understand this question.  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! Intern  Joined: 18 Mar 2013
Posts: 1
Re: If Sn is the sum of the first n terms of a certain sequence  [#permalink]

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2
1
Approach: Back Solve

Steps:
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

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Re: If Sn is the sum of the first n terms of a certain sequence  [#permalink]

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4
1
joseph0alexander wrote:
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

Hi All,

I'm totally lost and unable to understand this question.  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! 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?
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Re: If Sn is the sum of the first n terms of a certain sequence  [#permalink]

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Bunuel wrote:
Basically to get the sum of the first n terms we simply should substitute the value of n in the formula.

Okay I get it now. We're taking the first term of the sequence to be 1, because the question states that Sn = n(n+1) for all positive values of n or the first n terms of the sequence. So the number which is best qualified to be the first term of the sequence is the lowest positive number i.e. 1, and the equation must be correct when we take n=1 and any increment to it.

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Re: If Sn is the sum of the first n terms of a certain sequence  [#permalink]

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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.

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Re: If Sn is the sum of the first n terms of a certain sequence  [#permalink]

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1
angelfire213 wrote:
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).

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

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If Sn is the sum of the first n terms of a certain sequence  [#permalink]

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3
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
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Re: If Sn is the sum of the first n terms of a certain sequence  [#permalink]

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1
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)

manpreetsingh86 wrote:
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
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Re: If Sn is the sum of the first n terms of a certain sequence  [#permalink]

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angelfire213 wrote:
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)

manpreetsingh86 wrote:
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

No. S(2) would be the SUM of the first two terms in the sequence. We do not know what the sequence is. The sum is also represented by n(n+1) (eg. Sum of sequence = Sn = n(n+1)). Therefore, S2 = the sum of the first two numbers.

Makes sense?
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Re: If Sn is the sum of the first n terms of a certain sequence  [#permalink]

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MensaNumber wrote:
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

Sum of the first 'n' natural numbers=1/2 *n(n+1)
That implies sum of the first 'n' even numbers =n(n+1)
(2+4+6+8+....2n)=2(1+2+3+4+....+n)
Therefore, the third term of sequence is 6
The correct option is C
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If Sn is the sum of the first n terms of a certain sequence  [#permalink]

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This question basically asks whether you know the shortcut formula for the sum of the first positive even integers which is n(n+1). So from the notation that Sn is the sum of ALL positives we derive that the first then should be 2, second - 4 and third even - is 6 and so on
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Re: If Sn is the sum of the first n terms of a certain sequence  [#permalink]

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1
1
NoHalfMeasures wrote:
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

let A1, A2, A3 be the first 3 terms
Sum of first 2 terms = A1+A2 = 2(2+1) = 6 ..............(i)
Sum of the three terms = A1+A2+A3 = 3(3+1) = 12 ......(ii)

(ii)-(i) = A3= 6
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Re: If Sn is the sum of the first n terms of a certain sequence  [#permalink]

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NoHalfMeasures wrote:
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

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.

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Re: If Sn is the sum of the first n terms of a certain sequence  [#permalink]

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NoHalfMeasures wrote:
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

third term=sum of 3 terms minus sum of 2 terms
3(3+1)-2(2+1)=12-6=6
C
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Re: If Sn is the sum of the first n terms of a certain sequence  [#permalink]

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As I understand it

Sn= sum of values all terms upto n
Sn-1= sum of values of all terms upto n-1 therefore, does not include value of n.
therefore value of n term= Sn-(Sn-1)

if Sn= n(n+1)
then to find the value of 3rd term if we find the sum of first three term (upto 3rd term) and sum of first two terms (upto 2nd term) and then subtract it (S3-S2) which will give us the value of the 3rd term.
S3=3(3+1)=12 n=3
S2=2(2+1)=6 n=2
S3-S2= value of third term = 12-6=6
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Re: If Sn is the sum of the first n terms of a certain sequence  [#permalink]

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1
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
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