Math : Sequences & Progressions

"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

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

Are geometric and harmonic progressions tested/ important for GMAT?

I have just studied arithmetic progressions till now.

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

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

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

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."

Sequences & Progressions

This post is a part of [GMAT MATH BOOK]

created by: shrouded1



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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,...} \(a_1=-0.1, d=-1\)

Example 4
For every integer k from 1 to 10, inclusive, the kth term of a certain sequence is given by (-1)^(k+1) *(1/2^k). If T is the sum of the first 10 terms in the sequence then T is

a)greater than 2
b)between 1 and 2
c)between 1/2 and 1
d)between 1/4 and 1/2
e)less than 1/4.

Solution
The sequence given has first term 1/2 and each subsequent term can be obtained by multiplying with -1/2. So it is a GP. We can use the GP summation formula
\(S=b\frac{1-r^n}{1-r}=\frac{1}{2} * \frac{1-(-1/2)^{10}}{1-(-1/2)} = \frac{1}{3} * \frac{1023}{1024}\)
1023/1024 is very close to 1, so this sum is very close to 1/3
Answer is d

Example 5
The sum of the fourth and twelfth term of an arithmetic progression is 20. What is the sum of the first 15 terms of the arithmetic progression?
A. 300
B. 120
C. 150
D. 170
E. 270

Solution
\(a_4+a_12=20\)
\(a_4=a_1+3d, a_12=a_1+11d\)
\(2a_1+14d=20\)
Now we need the sum of first 15 terms, which is given by :
\(\frac{15}{2} (2a_1 + (15-1)d) = \frac{15}{2} * (2a_1+14d) = 150\)
Answer is (c)


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