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Sequences and Series Formula
Sequence and Series are one of the most important and basic concepts of arithmetic. A sequence is an arrangement of a group or set of objects in a particular order followed by some rules and regulations. If A1, A2, A3…., etc. denotes the term of sequence and 1,2,3,4…. denotes the position of term. The sequence can be defined based on the number of terms that can either be finite or infinite.
There is A1, A2, A3…An
Then Sn = A1+A2+A3+…. +An
A series can be defined as the sum of all terms of sequences. However, there must be a connection between all terms of sequences.
Different types of sequences
Arithmetic Sequences: This is a type of sequence in which every term is formed either by subtracting or adding a definite number to the preceding number.
Geometric Sequences: A sequence obtained by multiplying and dividing the definite number with the preceding number.
Harmonic Sequence: In this sequence, reciprocals of all the elements of the sequence create an arithmetic sequence.
Fibonacci Numbers: In this type, the elements are obtained by adding two preceding elements, and also the sequence begins with 0 and 1.
Type | Formula |
Sequence | a+1, a+2d… a+(n-1) d |
Common Difference | Common Ratio = r = ar(n-1)/ar(n-2) |
General Term | an = a + (n-1) d |
nth term from the last term | an = 1 – (n-1) d |
Sum of first n terms | Sn = n/2 (2a + (n-1) d) |
Solved Examples
Example 1:
If 4,7,10,13,16,19,22…. is a sequence, then find?
Common Difference
nth term
21st term
Common Difference= 7-4 = 3
The nth term is an arithmetic Sequence denoted as Tn= a +(n-1) d
Tn = 4+(n-1)3 = 3n+1
21st term = T21 = 4(21-1)3 = 64
Example 2:
Find the value of the 25th term of arithmetic Sequence 5,9,13,17…
Given, arithmetic sequences 5,9,13,17
For the 25th term, substitute n = 25
The first term a = 5
T25 = a + 24d = 101
