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Math : Sequences & Progressions

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Re: Math : Sequences & Progressions  [#permalink]

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New post 30 Jun 2014, 11:13
f an infinite GP is summable (|r|<1) then the sum is \frac{b_1}{1-r}

can someone please explain what these means with numbers?

i guess what i'm asking for here is a question where we would use this concept
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New post 30 Jun 2014, 11:53
sagnik2422 wrote:
f an infinite GP is summable (|r|<1) then the sum is \frac{b_1}{1-r}

can someone please explain what these means with numbers?

i guess what i'm asking for here is a question where we would use this concept


The sum of infinite geometric progression with common ratio \(|r|<1\), is \(sum=\frac{b}{1-r}\), where \(b\) is the first term.

For example, the sum of infinite geometric progression 1/2, 1/6, 1/18, 1/54, ... (first term = 1/2, common ratio = 1/3) is \(sum=\frac{b}{1-r}=\frac{\frac{1}{2}}{1-\frac{1}{3}}=\frac{3}{4}\).

Questions to practice:
for-every-integer-m-from-1-to-100-inclusive-the-mth-term-128575.html
for-every-integer-k-from-1-to-10-inclusive-the-kth-term-of-88874.html
a-square-is-drawn-by-joining-the-midpoints-of-the-sides-of-a-102880.html
m17q5-72268.html
ax-y-is-an-operation-that-adds-1-to-y-135277.html

Hope it helps.
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New post 09 Aug 2014, 20:24
"Summation
The sum of an infinite GP will be finite if absolute value of r < 1
The general sum of a n term GP with common ratio r is given by \(b_1*\frac{r^n - 1}{r-1}\)
If an infinite GP is summable (|r|<1) then the sum is \(\frac{b_1}{1-r}\)
"

Awesome post! After 4 years, I still want to read your post. Just need to correct the typo I hightlight than your post is perfect. The correct formula is
\(S=b_1\frac{1-r^n}{1-r}\)
not

\(b_1*\frac{r^n - 1}{r-1}\)
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New post 12 Aug 2014, 01:43
linhntle wrote:
"Summation
The sum of an infinite GP will be finite if absolute value of r < 1
The general sum of a n term GP with common ratio r is given by \(b_1*\frac{r^n - 1}{r-1}\)
If an infinite GP is summable (|r|<1) then the sum is \(\frac{b_1}{1-r}\)
"

Awesome post! After 4 years, I still want to read your post. Just need to correct the typo I hightlight than your post is perfect. The correct formula is
\(S=b_1\frac{1-r^n}{1-r}\)
not

\(b_1*\frac{r^n - 1}{r-1}\)


There is no typo there. Those two are the same: factor -1 from denominator and numerator and reduce.
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Re: Math : Sequences & Progressions  [#permalink]

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New post 26 Nov 2014, 03:56
One small correction as highlighted in bold:
In a finite AP, the mean of all the terms is equal to the mean of the middle two terms if n is even and the middle term if n is ODD. In either case this is also equal to the mean of the first and last terms first and last terms
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Re: Math : Sequences & Progressions  [#permalink]

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New post 24 Jul 2015, 03:03
Hello,

In "summation" for the arithmetic progression, where you give the general sum of a n term AP with common difference d, you have "a" instead of "a1", right?
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New post 29 Dec 2016, 22:45
Correction please.

..."n a finite AP, the mean of all the terms is equal to the mean of the middle two terms if n is even and the middle term if n is odd. In either case this is also equal to the mean of the first and last terms"
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New post 18 Apr 2017, 01:21
Are geometric and harmonic progressions tested/ important for GMAT?

I have just studied arithmetic progressions till now.
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New post 18 Apr 2017, 01:24
Shiv2016 wrote:
Are geometric and harmonic progressions tested/ important for GMAT?

I have just studied arithmetic progressions till now.


I would suggest have the basic knowledge of these as well. You may encounter a question on these if you are on your way to Q51. :-D
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Re: Math : Sequences & Progressions  [#permalink]

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New post 19 Aug 2017, 12:00
In the Geometric Progressions section under the Defining Properties-

Each of the following is necessary & sufficient for a sequence to be an AP :
..

It should be GP.

A similar typo is repeated in the Harmonic Progressions section under the Defining Properties. Must be a ctrl c + ctrl v issue :think:

Kindly correct the typo. Bunuel Thank you very much for the post.
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New post 20 Aug 2017, 01:28
shoumkrish wrote:
In the Geometric Progressions section under the Defining Properties-

Each of the following is necessary & sufficient for a sequence to be an AP :
..

It should be GP.

A similar typo is repeated in the Harmonic Progressions section under the Defining Properties. Must be a ctrl c + ctrl v issue :think:

Kindly correct the typo. Bunuel Thank you very much for the post.


Edited. Thank you.
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New post 30 Aug 2017, 23:49
Quote:
Misc Notes
A subsequence (any set of consequutive terms) of an AP is an AP

A subsequence (any set of consequutive terms) of a GP is a GP

A subsequence (any set of consequutive terms) of a HP is a HP

If given an AP, and I pick out a subsequence from that AP, consisting of the terms \(a_{i1},a_{i2},a_{i3},...\) such that \(i1,i2,i3\) are in AP then the new subsequence will also be an AP

For Example : Consider the AP with \(a_1=1, d=2\) {1,3,5,7,9,11,...}, so a_n=1+2*(n-1)=2n-1
Pick out the subsequence of terms \(a_5,a_{10},a_{15},...\)
New sequence is {9,19,29,...} which is an AP with \(a_1=9\) and \(d=10\)

If given a GP, and I pick out a subsequence from that GP, consisting of the terms \(b_{i1},b_{i2},b_{i3},...\) such that \(i1,i2,i3\) are in AP then the new subsequence will also be a GP


For Example : Consider the GP with \(b_1=1, r=2\) {1,2,4,8,16,32,...}, so b_n=2^(n-1)
Pick out the subsequence of terms \(b_2,b_4,b_6,...\)
New sequence is {4,16,64,...} which is a GP with \(b_1=4\) and \(r=4\)

The special sequence in which each term is the sum of previous two terms is known as the fibonacci sequence. It is neither an AP nor a GP. The first two terms are 1. {1,1,2,3,5,8,13,...}

In a finite AP, the mean of all the terms is equal to the mean of the middle two terms if n is even and the middle term if n is even. In either case this is also equal to the mean of the first and last terms





I am sure you mean that " In a finite AP, the mean of all the terms is equal to the mean of the middle two terms if n is even and the middle term if n is ODD. In either case this is also equal to the mean of the first and last terms "


Thank you so much for the post though, is very valuable :)
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New post 25 Nov 2017, 22:36
Very helpful post indeed. I have studied lots of concepts using different prep sources and I must say what I learn on here on the forum is priceless. Sometimes I feel like what the heck did I study or how did I miss that such an elegant solution exists. Many thanks!
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New post 10 Dec 2017, 07:09
1
Minor correction:
It should be "a finite AP, the mean of all the terms is also equal to the mean of the middle two terms if n is even and the middle term if n is odd."
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New post 10 Dec 2017, 07:49
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New post 28 Feb 2018, 12:00
shrouded1 wrote:
Sequences & Progressions
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Definition

Sequence : It is an ordered list of objects. It can be finite or infinite. The elements may repeat themselves more than once in the sequence, and their ordering is important unlike a set

Arithmetic Progressions


Definition
It is a special type of sequence in which the difference between successive terms is constant.

General Term
\(a_n = a_{n-1} + d = a_1 + (n-1)d\)
\(a_i\) is the ith term
\(d\) is the common difference
\(a_1\) is the first term

Defining Properties
Each of the following is necessary & sufficient for a sequence to be an AP :
  • \(a_i - a_{i-1} =\) Constant
  • If you pick any 3 consecutive terms, the middle one is the mean of the other two
  • For all i,j > k >= 1 : \(\frac{a_i - a_k}{i-k} = \frac{a_j-a_k}{j-k}\)

Summation
The sum of an infinite AP can never be finite except if \(a_1=0\) & \(d=0\)
The general sum of a n term AP with common difference d is given by \(\frac{n}{2}(2a+(n-1)d)\)
The sum formula may be re-written as \(n * Avg(a_1,a_n) = \frac{n}{2} * (FirstTerm+LastTerm)\)

Examples
  1. All odd positive integers : {1,3,5,7,...} \(a_1=1, d=2\)
  2. All positive multiples of 23 : {23,46,69,92,...} \(a_1=23, d=23\)
  3. All negative reals with decimal part 0.1 : {-0.1,-1.1,-2.1,-3.1,...} \(a_1=-0.1, d=-1\)




Bunuel can you help me with some questions. it is great post just lacks some examples in order to grasp the basic concept / more detailed explanation... can you please answer my questions below in red ? :)


\(a_n = a_{n-1} + d = a_1 + (n-1)d\) so \(a_{n-1}\) is the first term and it is the same as \(a_1\) ? \(a_n = a_{n-1} + d = a_1 + (n-1)d\)

\(a_i\) is the ith term (where do you guys see \(a_i\) / ith term in the above formula? :?
\(d\) is the common difference
\(a_1\) is the first term


Defining Properties
Each of the following is necessary & sufficient for a sequence to be an AP :
  • \(a_i - a_{i-1} =\) Constant <--- what does it mean ? could some give an example with real numbers :)
  • If you pick any 3 consecutive terms, the middle one is the mean of the other two
  • For all i,j > k >= 1 : \(\frac{a_i - a_k}{i-k} = \frac{a_j-a_k}{j-k}\)
<--- what does it mean ? could some give an example with real numbers:)
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Re: Math : Sequences & Progressions &nbs [#permalink] 28 Feb 2018, 12:00

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