Arithmetic sequence

Arithmetic Sequence:

An arithmetic sequence is defined as a group of numbers in which any two consecutive or successive terms have a common difference.
Example:
In the sequence of following numbers:
3, 7, 11, 15, 19, ....
the common difference between any two consecutive numbers is 4.
Therefore, the numbers 3, 7, 11, 15, 19 ...are said to be in Arithmetic Sequence
Denotation of terms in an Arithmetic Sequence
In an Arithmetic Sequence, the following denotations are generally followed:
a denotes the first term,
d, the common difference and
n, the position of any term or a particular term.
Then the n terms of an Arithmetic Sequence are written as
a, a +d, a + 2d, a + 3d, a + 4d, ............., a + (n – 1)×d
In the above sequence of n terms
a is 1^st term, 
a + d is 2^nd term,
a + 2d is 3^rd term, and
a + (n – 1) ×d is the last term
a + (n – 1) × d is also called the last term or the n^th  term  or still the general term of the above arithmetic sequence.
We will discuss below the formulae for finding any particular term and sum of any number of terms in an arithmetic sequence.
What is n^th term of a sequence?
The nth term of a sequence denotes the position of a term or a particular term in the sequence.
Any term that is to be found is referred to as n^th term or general term.
How to find the nth term/general term of an Arithmetic Sequence?
Let the n terms of an Arithmetic Sequence with a common difference ‘d’ be
a, a + d, a +2d, ........ a + (n – 1) d
Any term in an Arithmetic Sequence is called general term or nth term and it is denoted as an
To find this aⁿ the following formula is applied:
a_n = a + (n – 1) d,  where
a is the first term,
d is the common difference and
n is the position of the term.
Example
1. The first three terms of an Arithmetic Sequence are 3, 7, 11 ...What is the 50^th term?
Solution:
In the A.S. 3, 7, 11, the common difference is 4.
To find the 50th term, apply the above formula for nth term
a_n = a + (n – 1)×d.
In this the first term a = 3, common difference d = 4 and n = 50.
Applying the above formula, the 50^th term is
3 + (50 – 1) × 4 = 3 + 196 = 199
2. In an arithmetic sequence with a common difference of 5, the 25^th term is 180. What is the 15^th term of this sequence?
Solution:
Represent 25^th term as a₂₅.
Next, from the nⁿ term formula for an A.S.:  a_n = a + (n – 1) ×d,
we can write, a₂₅ = 180, n = 25, d = 5.
What needs to be found is the first term, a.
Applying the nth term formula, we get
180 = a + (25 – 1) × 5,
180 = a + 120, so, a = 60.
Again, from the nth term formula, the 15^th term is
a₁₅ = 60 + (15 – 1) × 5 = 130

How to find the sum of n terms of an Arithmetic Sequence:
Let the n terms of an Arithmetic Sequence be
a, a + d, a + 2d, ............, a + (n – 1)× d
If S_n is the sum of the above n terms which are in Arithmetic Sequence, then
S_n  = n/2 [2a + (n – 1 ) d]
Now, let ‘l’ denote the last term, then
Since l = a + (n – 1)d, therefore
S_n = n/2[a + l]
Example:
1. The terms of an Arithmetic Sequence are 2, 5, 8, 11, ..........
What is the sum of the first 51 terms of this sequence?
Solution:
Now, first term, a = 2,
Common difference, d = 3, 
and the last term is
l = a + (n – 1) × d is
2 + (51 – 1) × 3 = 2 + 150 = 152
Therefore, S = 51/2 [2 + 152] = 3927

Geometric sequence.

Sequences and series.

Arithmetic progression.

 

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