Summary
- When the terms of a sequence are added together a series is formed.
- A series can be denoted by a sigma notation where a is the first value of n terms, b is the last value of the n terms and x is the expression of the given sequence.
- The formula for n-th term of an arithmetic progression is:
- Sum of the series is given by the formula:
- The formula for n-th term of a geometric progression is:
- Formula to work out the sum of geometric progression:
- Sum to infinity of the geometric progression:
Let’s just quickly recall that sequence is a set of numbers in a given order with a rule for obtaining the terms. Moving on, when the terms of a sequence are added together a series is formed.
For example:
Sequence | Series |
---|---|
{1, 3, 5, 7….} | 1 + 3 + 5 + 7 + .... |
{5, 10, 15, 20….} | 5 + 10 + 15 + 20 + .... |
{10, 100, 1000,.....} | 10 + 100 + 1000 +..... |
A series can be denoted by a sigma notation where a is the first value of n terms, b is the last value of the n terms and x is the expression of the given sequence. This sigma notation is a Greek capital and is used to represent a sum.
Let’s consider we are given a series 2 + 4 + 6 + 8 + 10.
We can see that n = 5 is the last value of n and the expression of the sequence would be .
Therefore, we can write it using the sigma notation in the form:
Next, to find the sum of the above sequence, we carry out the following steps:
Furthermore, in this article we will study series in the following types of sequences.
Arithmetic Progression
An arithmetic progression is a sequence of numbers which increases by a constant amount which could be either positive or negative. This amount is called the ”common difference” (d) and the starting number is called ”first term” (a).
In general, if an arithmetic progression has first term a and common difference d, then in standard form, it is written as:
So the formula for n-th term of an arithmetic progression is:
Here: a = first term, d = common difference, n = no. of terms, = n-th term
Example #1
Q. Find the 20 th term of the sequence 5, 8, 11, 14, 17,…..
Solution:
We know a = 5, d = 8 – 5 = 3 also, we know it is an arithmetic sequence
Ans
Sum of Arithmetic Progression
Suppose we are asked to find the sum of first 100 numbers in an arithmetic progression. Adding all the numbers one by one will be a very tedious job. Gauss, a german mathematics solved this problem in a very simple manner.
If the first term is a, and the last term is l, the number of terms is n, then the sum of the series is given by the formula:
Or
Example #2
Q. Find the sum of the first 10 terms of the sequence 5, 8, 11, 14, 17,…..
Solution:
We know a = 5, d = 3 and n = 10. We know it is an arithmetic sequence.
Ans
Geometric Progression
Geometric progression is a sequence in which each term is multiplied by a constant r known as the common ratio. Thus if a is the first term, r is the common ratio, then the standard form of a geometric progression is:
Hence to find the term of the geometric progression, we use the formula:
Where r can be worked out by dividing two consecutive terms in such a way:
Example #3
Q. Find the 10th term of the geometric progression 3, 6, 12,….
Solution:
We know a = 3, n = 10 and we can find r
Use the above formula to work out the 10th term.
Ans
Sum of geometric progression
Just like we studied a formula to calculate the sum of arithmetic progression. Similarly, we are given a formula to work out the sum of geometric progression:
Here a = first term, r = common ratio, n = no. of terms
Sum to infinity of a geometric progression
Consider the following infinite geometric progression:
We can see that a = 1 and ,
As n increases decreases, when n becomes very large i.e , becomes very small and . The value of becomes .
We call the limiting value of as the sum to infinity of the geometric progression.
Therefore, if we consider -1 < r < 1, then when .
Hence the formula becomes:
Example #4
Q. Find the sum to infinity of the geometric progression:
Solution:
a = 2 and
Using the above formula:
Ans