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# If S is the infinite sequence S1 = 6, S2 = 12, ..., Sn =Sn-1

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If S is the infinite sequence S1 = 6, S2 = 12, ..., Sn =Sn-1 [#permalink]  04 Oct 2010, 07:17
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If S is the infinite sequence S1 = 6, S2 = 12, ..., Sn = Sn-1 + 6,..., what is the sum of all terms in the set {S13, S14, ..., S28}?

a) 1,800

b) 1,845

c) 1,890

d) 1,968

e) 2,016
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Re: s in infinite sequence [#permalink]  04 Oct 2010, 07:51
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anilnandyala wrote:
If S is the infinite sequence S1 = 6, S2 = 12, ..., Sn = Sn-1 + 6,..., what is the sum of all terms in the set {S13, S14, ..., S28}?

a) 1,800

b) 1,845

c) 1,890

d) 1,968

e) 2,016

Given: s_1=6 and s_n=s_{n-1}+6=s_1+6(n-1).

Question: sum of 16 elements from this sequence s_{13}+s_{14}+...+s_{28}=?

As s_n=s_1+6(n-1) then s_{13}=6+6(13-1)=78 and s_{28}=6+6(28-1)=168.

Sum of 16 evenly spaced terms would be \frac{first \ term+last \ term}{2}*# \ of \ terms=\frac{s_{13}+s_{28}}{2}*16=\frac{78+168}{2}*16=1968.

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Re: s in infinite sequence [#permalink]  04 Oct 2010, 11:50
S1 = 6, S2 = 12, ..., Sn = Sn-1 + 6,..., what is the sum of all terms in the set {S13, S14, ..., S28}?

formula = n/2(firstterm + last term)

= s13 to s28 ---> we have 16 terms so n will be = 16
first term = s13 = since term is getting added 6 to the next term, the 13th term will be = 13*6 = 78
s28 = 28*6 = 168

so the sum = n/2(first term + last term) = = > 16/2(78+168) ====> 1968
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Re: s in infinite sequence [#permalink]  04 Oct 2010, 12:38
Sn = 6*n...
S1 = 6 * 1; S2 = 6*2; ...

From S13 to S28: 6*13 + 6*14 + ... + 6*28 = 6* (13 + 14 + ... + 28)

13 + 14 + ...= 16 * (13+28)/2 = 328

Therefore 6 * 328 = 1968 (only term that ends with 8)
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Re: s in infinite sequence [#permalink]  04 Oct 2010, 17:49
s1=6, s2=12.
so s13=13*6 and s28=28*6
so the sum = number of terms*(first term + last term)/2
= 16*6*(13+28)/2
=1968
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Re: s in infinite sequence [#permalink]  06 Oct 2010, 05:17
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are we supposed to know this formula for GMAT?
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Sequence [#permalink]  13 Nov 2010, 17:59
This question comes from Manhattan GMAT. I don't understand how you find the rule for this sequence. I just used the "S_n=S_(n-1)+6" to try to find the numbers in the sequence, but I was wrong. It's 6n. Once I see that that is the answer in the solution I can see it, but how can I arrive to that on my own?

If S is the infinite sequence S_1 = 6, S_2 = 12, ..., S_n = S_(n-1) + 6,..., what is the sum of all terms in the set {S_13, S_14, ..., S_28}?
a) 1,800
b) 1,845
c) 1,890
d) 1,968
e) 2,016
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Re: Sequence [#permalink]  13 Nov 2010, 18:59
MateoLibre wrote:
This question comes from Manhattan GMAT. I don't understand how you find the rule for this sequence. I just used the "S_n=S_(n-1)+6" to try to find the numbers in the sequence, but I was wrong. It's 6n. Once I see that that is the answer in the solution I can see it, but how can I arrive to that on my own?

If S is the infinite sequence S_1 = 6, S_2 = 12, ..., S_n = S_(n-1) + 6,..., what is the sum of all terms in the set {S_{13}, S_{14}, ..., S_{28}}?
a) 1,800
b) 1,845
c) 1,890
d) 1,968
e) 2,016

This is an arithmetic progression: 6, 12, 18, 24, 30...... (or I can say it is the multiplication table of 6)
When they say S(n) = S(n - 1) + 6, they are giving you that every subsequent term is 6 more but just writing down the first few numbers you will realize that it is just the table of 6. This happens because the first term is 6 so every time you add 6, it just becomes the next number in the multiplication table of 6. How will you learn to observe such things? Just by practice!

First term - 6
Second term - 6x2
Third term - 6x3 and so on
so 13th term will be 6x13
14th term will be 6x14
.
.
28th term will be 6x28
I need to add 6x13 + 6x14 +....6x28 = 6(13 + 14 + ...28)
13 + 14 +..28 = Sum of first 28 terms - Sum of first 12 terms = \frac{28*29}{2} - \frac{12*13}{2} = 1968
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Re: Sequence [#permalink]  13 Nov 2010, 22:25
need to add 6x13 + 6x14 +....6x28 = 6(13 + 14 + ...28)
so 6 * (13+28)/2 * 16 = 3 * 41 * 16 = 1968

(sum of an arithmatic sequence = (first term + last term)/2 * no of terms)
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Re: s in infinite sequence [#permalink]  14 Nov 2010, 04:26
prashantbacchewar wrote:
are we supposed to know this formula for GMAT?

The formula is simply the formula of the sum of an AP.
If a is the first term, d is the common difference, and n is the number of terms, then

S = \frac{n}{2}(2a + (n-1)d)
or
S = \frac{n}{2}(a + b)
b is the last term of the progression which is written as a + (n-1)d.
The logic behind it is that take the average of the AP which is (a + b)/2 and multiply it by n, the number of terms as if the average in added n times rather than individual numbers. It makes complete sense. Look at the example:

AP with 3 terms: 4 7 10
7 is the average. 4 is 3 less than 7 and 10 is 3 more. Rather than adding 4 and 10 to 7, I can add 7 two more times and still get the same answer.

Since GMAT does not focus on formulas, generally you can solve the question in other ways too (like I have done in my solution).
Of course some basic formulas you should be good with and Sum of n consecutive terms starting from 1 = n(n + 1)/2 is one of them.
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Re: s in infinite sequence [#permalink]  25 Mar 2011, 04:40
There is a much easier way to deal with this problem.
a)13th member is equal to 13*6 = smth8 (ok, 78, but 7 does not matter)
b) how many members are there between 28th and 14th members (i.e how many members will we add to 13th member?) = (28-14)+1 = 15 member.
c) 15*6 = smth0 That is important: the sum of members 14th-28th will has the unit digit ZERO!
d) now... smth8+smth0 = smth8 or answer D in that case
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Re: s in infinite sequence [#permalink]  27 Mar 2011, 01:39
(2)^1/3, (5)^1/6, (10)^1/10, (30)^1/15

s13 = s1 + (12) * 6

=> s13 = 13 * 6 = 78

s28 = s1 + 27 * 6

s28 = 28 * 6 = 168

So Sum = 16 * (78 + 168)/2

= 8 * 246

The answer must end with last digit as 8 and We can stop multiplying here as there is only 1 answer like that.

= 1968

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Re: s in infinite sequence [#permalink]  15 Feb 2012, 06:50
Can we solve the below sum using this approach

S13-S28 = {S1-S28} - {S1-S13}

S1-S28 = n/2 {2a+(n-1)d} =28/2 { 2*6 + 27*6} = 2436
S1-S13 = n/2{2a+(n-1)d} = 13/2{2*6+12 *6} = 546

S13-S28 =2436-546 =1890

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Re: s in infinite sequence [#permalink]  15 Feb 2012, 11:19
prakarp wrote:
Can we solve the below sum using this approach

S13-S28 = {S1-S28} - {S1-S13}

S1-S28 = n/2 {2a+(n-1)d} =28/2 { 2*6 + 27*6} = 2436
S1-S13 = n/2{2a+(n-1)d} = 13/2{2*6+12 *6} = 546

S13-S28 =2436-546 =1890

The sum of all terms in the set {S13, S14, ..., S28} means the sum of all the terms from S13 to S28, inclusive. So, it equals to the sum of first 28 terms minus the sum of first 12 terms;

Hence it should be: the sum of first 28 terms minus the sum of first 12 terms = 2436-468=1968.

Hope it's clear.
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Re: s in infinite sequence   [#permalink] 15 Feb 2012, 11:19
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