Sequence and Series:
In this article, read all you need to know about sequence and series.
An arithmetic sequence is a sequence of numbers such that the difference of any two successive members of the sequence is a constant.
2,4,6,8,10….is an arithmetic sequence with the common difference 2.
A sequence may be named or referred to by an upper-case letter such as “A” or “S“. The terms of a sequence are usually named something like “ai” or “an“, with the sub-scripted letter “i” or “n” being the “index” or the counter. So the second term of a sequence might be named “a2“, and “a12” would designate the twelfth term.
If the first term of an arithmetic sequence is a1 and the common difference is d, then the nth term of the sequence is given by:
And, sum of any arithmetic series is: Sn = (n/2) [2a + (n-1)d]
- A series in which the difference between any term and its previous term is constant (eg. 2 + 5 + 8 + 11 + 14) is an arithmetic series.
- A sequence is said to be an arithmetic progression if there exists a number, say such that , = . The constant number is called the common difference of the corresponding
- If and be the first term and common difference of the , then T_n is said to be the last term of the series and the formulafor calculation is = , where .
- If . and be the first term and common difference of the . , then the sum of first n terms, is given by =
- T1 = and for , we have =
- If the sequence is an then the numbers are called the arithmetic means between and .
- The . between given numbers and is equal to (
- The sum of n between given numbers and is equal to times the between and .
- If are in then for any are in are in are in are in
- If the sum of three numbers in is given, then the numbers should be taken as If the sum of four numbers in is given, then the numbers should be taken as
Example 1: Find 10th term in the series 1, 3, 5, 7, …
a = 1
d = 3 – 1 = 2
10th term, t10 = a + (n-1)d = 1 + (10 – 1)2 = 1 + 18 = 19
Example 2: Find 16th term in the series 7, 13, 19, 25, …
a = 7
d = 13 – 7 = 6
16th term, t16 = a + (n-1)d = 7 + (16 – 1)6 = 7 + 90 = 97
Example 3: Find the number of terms in the series 8, 12, 16, . . .72
a = 8
l = 72
d = 12 – 8 = 4
Example 4: Find 4 + 7 + 10 + 13 + 16 + . . . up to 20 terms
a = 4
d = 7 – 4 = 3
Sum of first 20 terms, S20
Example 5: Find 6 + 9 + 12 + . . . + 30
a = 6
l = 30
d = 9 – 6 = 3
If a, b, c are in AP, b is the Arithmetic Mean (AM) between a and c. In this case,
If a, a1, a2 … an, b are in AP we can say that a1, a2 … an are the n Arithmetic Means between a and b.
To solve most of the problems related to AP, the terms can be conveniently taken as
3 terms: (a – d), a, (a +d)
4 terms: (a – 3d), (a – d), (a + d), (a +3d)
5 terms: (a – 2d), (a – d), a, (a + d), (a +2d)
In an AP, sum of terms equidistant from beginning and end will be constant.
Non-zero numbersare in Harmonic Progression(HP) if are in AP. Harmonic Progression is also known as harmonic sequence.
Three non-zero numbers a, b, c will be in HP, ifare in AP
The Harmonic Mean(HM) between two numbers a and b =
If a, b, c are in HP,
Geometric Progression(GP) or Geometric Sequence is sequence of non-zero numbers in which the ratio of any term and its preceding term is always constant.
A geometric progression(GP) is given by a, ar, ar2, ar3, … [where a = the first term , r = the common ratio]
Sum of an Infinite Geometric Progression
where a= the first term , r = common ratio
a = 1, r =
Here -1 < r < 1. Hence,
Power Series – Important Formulas