Sequence, Series and Progressions

May 12, 2015

Sequence, Series and Progressions

We will begin by defining a few common terms.

Sequences are a set of numbers, which are arranged according to any specific rule. There is no exception for any type of numbers, any type of rules according to which they are arranged. The set of numbers should have a definite, logical rule according to which they are arranged. It need not be a mathematical formula, but it should be logical. Such a set of numbers are called a sequence of numbers.

For example, the following is a sequence of numbers, because they are arranged according to a definite rule:

{2, 4, 6, 8, 10, 12} Rule: nth term = 2n

The following is a sequence of odd numbers:

{3, 5, 7, 9, 11, 13} Rule: nth term = 2n + 1

The following is also a sequence of numbers, as they too have a logical rule:

{2, 3, 5, 7, 11, 13, 17} Rule: Prime numbers

A series, on the other hand, is a sequence of numbers that is added by + signs. The term 'series' is closely related to the total sum of a sequence of numbers. However, the word 'series' is said to represent the sum of the numbers, and not the sum itself. There is only one difference directly visible between a series and a sequence: The numbers in a series are separated by plus (+) signs, whereas the numbers in a sequence are separated by commas (,).

For example, the following is a series of numbers, because they are separated by + signs, and they are arranged according to a definite rule:

{2+4+6+8+10+12}

The following is a series associated with the sequence of odd numbers:

{3+5+7+9+11+13}

Another important characteristic of a series is that it is always based on a sequence. A series of numbers is always associated with a sequence of numbers.

Progressions are yet another type of number sets which are arranged according to some definite rule. The difference between a progression and a sequence is that a progression has a specific formula to calculate its nth term, whereas a sequence can be based on a logical rule like 'a group of prime numbers'.

Now you may be wondering that a set of prime numbers should be a progression because we can predict its nth term, but a progression needs a specifically stated formula, and, it is to be noted that prime numbers cannot be predicted with the help of any formula; Till date, the formula for the nth prime number has not be found. This means that we can only calculate the nth prime number with the method of selecting each successive number and checking whether it is prime or not.

Therefore {2, 4, 6, 8, 10, 12} represents a progression where the nth term is given by '2n', and, on the other hand, {2, 3, 5, 7, 11, 13, 17} represents a sequence that is not a progression, because although it is based on a definite logical rule (prime numbers), but there is no formula to calculate its nth term.

Now we will look into a few of the more important progressions for CAT and other entrance exams:

Arithmetic progression (A.P)

In mathematics, an arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15 … is an arithmetic progression with common difference of 2.

If the initial term of an arithmetic progression is a1 and the common difference of successive members is d, then the nth term of the sequence (an) is given by:

an = a1 + (n - 1)d

Also the sum of first n terms of the series is given by the formula:

Sn = (a1 + l) * n/ 2, where l is the last term of the series.

A few properties of A.P:

If a certain number is added or subtracted or multiplied or divided from all the terms of an A.P then the resulting sequence is also an A.P.
If there are 2 A.Ps then the sequence made by their addition or subtarction is also an A.P.
If you are to take an odd number of terms as A.P it is usually recommended you take m - k, m , m + k as the terms. For even series you can take, a - 3d, a - d, a + d, a + 2d.

Next coming to the concept of Arithmetic Mean(A.M)

simple A.M = ( a1 + a2 + … + an) / n

For odd number of terms it is usually the middle term, for even number of terms it goes like (a1 + an)/2 or (a2 + an-1)/2 and so on. A.M is related to Sn by the following relation: Sn = A.M * n.

A few properties of A.M:

If a certain number is added or subtracted or multiplied or divided from all the terms of an A.P then the the same will also happen with the A.M.

To insert a n A.Ms (m1 , m2 , …, mn) between 2 numbers a and b:

m1 = a + ((b - a)/(n + 1)) and so on.. where ((b - a)/(n + 1)) plays the role of d. This is easy to get as a is 1st number and b is the last as there will be n + 2( a & b) numbers in total so the number of d’s will be n+1. hence d = ((b - a)/(n + 1))

There is also another thing called weighted A.M :

If the values of x1, x2, x3 has got the w1, w2 , w3 weights assigned to them we have the weighted A.M as

xw = (x1w1 + x2w2 + x3w3) / (w1 + w2 + w3)

Note: the A.M is used to calculate the average speed when different distances are traveled in the same time.

Next we will move on to Geometric Progression(G.P):

here the terms have a common ratio between them like a, ar, ar2, ar3 and so on… the common ratio being given by r.

nth term of the sequence (an) is given by:

an = a1 * r(n - 1)

Also the sum of first n terms of the series is given by the formula:

Sn = (lr - a)/(r - 1), where l is the last term of the series and where r > 1, for r < 1 . Sn = (a - lr)/(1- r)

S= a /(1 - r), it is actually quite amazing that the sum an infinite series sums to be finite. The condition is that the series converges absolutely when the value of |r| < 1, we will not go into the details of convergence now. Now the sum of squares or cubes of the infinite series of numbers will be given by

a2 /(1 - r2) & a3 /(1 - r3) respectively.

Next moving on to the Geometric Mean(G.M),

G.M = (a1 * a2 * a3….an)1/n

To insert a n G.Ms (m1 , m2 , …, mn) between 2 numbers a and b:

m1 = a * (b / a)(n + 1) and so on.. where (b / a)(n + 1) plays the role of r, the explanation is the same as that of A.M.

A few properties of G.P:

If a certain number is multiplied or divided from all the terms of an G.P then the resulting series will also be a G.P.
If there are 2 G.Ps then the sequence made by their multiplication or divison is also a G.P.
If there are odd numbers in a G.P then the middle most term is the G.M, for even number of terms (a1 * an)1/2 or (a2 * an-1)1/2 and so on..
If the G.M of one set of a numbers is x and that of another set of b numbers is y, then the G.M of the combined set is (xa + yb)/(a + b)
If you are to take an odd number of terms as G.P it is usually recommended you take a/r, a, ar as the terms. For even series you can take a/r3, a/r, ar, ar3.

Now moving on to Harmonic Progression(H.P) , a series of numbers a, b, c , …. are said to be in H.P if their reciprocals 1/a, 1/b, 1/c, …. are in A.P or the vice versa. There are no specific formulas for this, all related questions are solved by changing it to A.P and making use of its corresponding properties.

There is also a H.M, for a series of numbers a1, a2, a3, …. an

H.M = n / (1/a1 + 1/a2 + 1/a3 + …. + 1/an), so for 2 numbers a and b the H.M is 2ab/(a + b).

If a set of weights w1, w2, ..wn is associated to the dataset x1, x2, ..xn the weighted harmonic mean is defined by

$\frac{\sum_{i=1}^n w_i }{ \sum_{i=1}^n \frac{w_i}{x_i}}.$

Note: the H.M is used to calculate the average speed when same distances are traveled in the different speed.

Note: the weighted H.M is used to calculate the average speed when different distances are traveled in the different speed.

Next coming to the last series Arithmetico Geometric Sequence:

this series is formed when an A.P series a1, a2, …. an and a G.P series b1, b2, …. bn are brought together to form the form the series

a1b1, a2b2, … so on. It is of the form ab, (a+d)br, (a+2d)br2, …

the series is given by the formula Sn = ab / (1-r) + dbr( 1-rn-1) / (1-r)2 - (a + (n-1)d)brn)/ (1-r)

S= ab / (1-r) + dbr / (1-r)2

Before wrapping up, A.M of 2 numbers is (a + b)/2, their G.M is a.b and their H.M is 2 / (a + b)

so the relation between the 3 means are given by GM2 = AM * HM and also AM > GM > HM

A few common sums are:

sum of first n natural numbers: 1, 2, ..n = n(n + 1)/2
sum of first n odd numbers: 1, 3, .. = n2
sum of first n even numbers: 2, 4, .. = n( n + 1)
sum of the squares of the first n natural numbers: 12, 22, …= n( n + 1) (2n + 1)/6
sum of the cubes of the first n natural numbers: 13, 23, …= {n ( n + 1 )/2}2

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Cat Kahini

Sequence, Series and Progressions

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