What are Sequence and Series?

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence.  It was invented by German mathematician Carl Friedrich Gauss.

How to Master Sequence and Series?

We will see some examples to have a better understanding of this. Suppose that in a large barn there are so many different kinds of fruits. In the first box, there is an apple. In the second, one finds an orange. A banana is found in the third, strawberry in the fourth, and again an apple in the fifth. Now if someone asks what is in the barn, one will say the collection of fruits that he or she has seen.

How to Distinguish Partial Sums, Series, and Sequence?

An apple, orange, banana, strawberry, and again an apple is the sequence. This sequence is finite and box 1 and 5 has apple. The terms can also repeat. Let us take the collection of natural numbers. This sequence is infinite. Now, how many fruits are there in the barn? How do we count? We associate each fruit with a natural number. So, the first fruit is apple, and it is called the first term of the sequence of fruits. It is denoted by a₁. The second fruit is the second term a₂ and so on. The expression we get by adding one term with the other term of the sequence is called a series and is written as a₁+a₂+a₃+…. The series can be finite or infinite and this depends on the terms we take from the sequence.

Partial Sum of Finite Series

How many fruits are there in the last four boxes?

The sum of each fruit gives 1+1+1+1=4. This is a series, and it is built from the terms that are there in the sequence. Here, we have added only a part of the sequence, the last four terms which are defined as the partial sum. So, 1+1+1+1 is a partial sum of the finite series 1+1+1+1+1.

Partial Sum of Infinite Series

An infinite series has no end. The sum of the first hundred terms of the odd natural numbers is an example of a partial sum of infinite series.

Algebraic Formula

If we take the sequence of even natural numbers it can be generally written by the algebraic formula 2n where n= 1, 2, 3,… and the collection includes the terms 2,4,6 and so on. What is the tenth term a10? Here n= 10, $a_{10}=2\times 10=20$. Find a15. But the sequence of composite numbers (1, 4, 6, 8, 9,…) cannot be represented by any formula and it has no pattern.

How are sequence and series used in real life? How to notice the number of petals in flowers? They follow some patterns although it is not easy to observe the pattern. It is called the Fibonacci sequence. The collection has the numbers 1, 1, 2, 3, 5, 8,… which has a rule that when you add one term in the sequence with its previous term you get the next term. Find out what term comes after 8 in this sequence.

Arithmetic Sequence

Consider the multiples of 2. The first five terms are 2, 4, 6, 8, and 10. Take some time and think about the common things that can be said about these five numbers. The difference between two terms which are next to each other is common and it is 2. It is denoted by d. When you add this common difference to one term, we get the next term. We write the 6th term if we know the 5th term by adding 2 to the 5th term. This can be generally written as a6=a5+d. Think about how we can write $a_{30}\left(a_{30}=a_{29}+2\right)$. If the first term is a, then the second term is a+ d, the third term is a+ d+ d which is a+2d, and so on. The 9th term is a+ 8d. If we know the first term a and the difference d, we can write any term. To find the nth term where n is any natural number, use the formula $a_n=a+\left(n-1\right)d$. We can find a30 without even knowing a29. Here n is 30.

$\begin{array}{l}{a}_n=a+\left(n-1\right)d\\ a_{30}=2+\left(30-1\right)2=60\end{array}$

If the last term is 80, put an=80 in $a_n=a+\left(n-1\right)d$.

In an arithmetic sequence, if one adds a constant term to each of the terms, it is still an arithmetic sequence. In the same way, we can remove, multiply or divide a constant from each term. The result will again be an arithmetic sequence.

Arithmetic Mean

Take 2 numbers a and b. Let a= 10 and b= 50. Can we find the middle number between 10 and 50? We can find the sum of 10 and 50 and divide the result by 2. 10+50=60 and dividing 60 by 2 gives 30. This is the formula to find the arithmetic mean. So, 30 is the middle term. If there are two packets of cocoa powder with one pack having 350 grams and the other pack having 150 grams, we can add the two packs together and divide the total by two. By doing this, we get two equal portions, one with 250 grams and the other with 250 grams using the arithmetic mean. Try inserting a number between 12 and 14 using an arithmetic mean.

Arithmetic Series

An arithmetic series is the sum of terms in an arithmetic sequence. If $\left\{a_k\right\}$ is an arithmetic sequence with the first term as ‘a’ and common difference as ‘d’, then the sum of the first n terms is evaluated by the formula $S_n=\frac{n}{2}\left[2a+\left(n-1\right)d\right]$. It is called the arithmetic series corresponding to the arithmetic sequence $\left\{a_k\right\}$.

Note that the nth term in an arithmetic series with first the term as ‘a’ and common difference as ‘d’ is written as $a_n=a+\left(n-1\right)d$, so the sum $S_n=\frac{n}{2}\left[2a+\left(n-1\right)d\right]$ can be simplified as follows:

$\begin{array}{c}{S}_n=\frac{n}{2}\left[2a+\left(n-1\right)d\right]\\ =\frac{n}{2}\left[a+a+\left(n-1\right)d\right]\\ =\frac{n}{2}\left[a+a_n\right]\end{array}$

This implies that the sum is n times the arithmetic mean of the first and last term.

Geometric Sequence

Consider the sequence 3, 6, 12, 24,… and find the relation between 3 and 6, 6 and 12, and so on. 3 is the first term a1. $\frac{6}{3}=2,\frac{12}{6}=2,\frac{24}{12}=2$ and so on. The ratio r between a term and its previous term is 2 in all the cases except the first term. This sequence is called a geometric sequence if all the terms are non-zero and the ratio r is the same. The first term is a, the second term, a2, is $a\times r$, the third term a3 is $a\times r^2$. What is a4? It is $a\times r^3$. Generally, any term n can be written as $a_n=a\times r^{n-1}$. Try finding a5 in this sequence.

$\begin{array}{l}{a}_n=a\times r^{n-1}\\ a_5=3\times 2^4=48\end{array}$

Geometric Mean

Take two numbers a and b. Let's take a= 2 and b= 8. Multiply 2 and 8 and take find the square root of it. This is the formula to find the geometric mean. We get, $2\times 8=16,\sqrt{16}=4$. This is the geometric mean of two numbers a and b and it lies between a and b. 4 is inserted between 2 and 8. We can insert many numbers in this way.

Geometric Series

A geometric series is the sum of terms in a geometric sequence. If $\left\{a_k\right\}$ is a geometric sequence with the first term as ‘a’ and the common ratio as ‘r’, then the geometric series is evaluated by the formula $S_n=a\left(\frac{1-r^n}{1-r}\right)$, if $r\ne 1$.

If $\left|r\right|<1$, then the geometric series approaches the value $\frac{a}{1-r}$ as the number of terms becomes infinitely large.

Arithmetic and Geometric Sequence

These two sequences are considered the simplest sequence to calculate.

Arithmetic sequence shifts from one term to the next by always adding or subtracting the same value. The number added or subtracted at each stage of the arithmetic sequence is called the ‘common difference’ because while subtracting the successive terms, the common value is always derived.

For instance,

2,4,6,8,10… is an arithmetic sequence because each step adds 2.

3, -1, -5, -9, is an arithmetic sequence because each step subtracts 4.

Geometric sequence shifts from one term to next by multiplying or dividing by the same value

For instance,

2,4,8,16, is a geometric sequence because each step multiplies by 2.

9,3,1,1/3, is a geometric sequence because each step is divided by 3.

Formulas

Common Mistakes

A series does not mean the sum of all the terms. It is just indicating the summing of the terms which we take. Do not confuse geometric sequence with the arithmetic sequence.

Context and Applications 

It is mostly asked in K-12, undergraduate, postgraduate entrance examinations.

This topic is significant in the professional exams for both undergraduate and graduate courses, especially for

• B.Sc. in Mathematics 
• M.Sc. in Mathematics