Arithmetic sequence formula is used to calculate the nth term of an arithmetic sequence. To recall, a sequence is an ordered list of numbers. The sum of the terms of a sequence is called a series. An arithmetic sequence or arithmetic progression is a sequence in which each term is created or obtained by adding or subtracting a common number to its preceding term or value. In other words, the difference between the adjacent terms in the arithmetic sequence is the same.
Formulas of Arithmetic Sequence
For the formulas of an arithmetic sequence, it is important to know the 1st term of the sequence, the number of terms and the common difference. Now,
| Arithmetic Sequence Formulas | |
|---|---|
| nth Term Formula | an = a1 + (n – 1)d |
| Sum of First n Terms | Sn = n/2 (first term + last term) |
Where,
- an = nth term that has to be found
- a1 = 1st term in the sequence
- n = Number of terms
- d = Common difference
- Sn = Sum of n terms
A few solved problems on the arithmetic sequence are given below.
Solved Examples Using Arithmetic Sequence Formula
Question 1: Find the 16th term in arithmetic sequence 0, 2, 4, 6, 8, 10, 12, 14…..
Solution:
The given arithmetic sequence is:
0, 2, 4, 6, 8, 10, 12, 14, …..
nth term formula is:
an = a1 + (n – 1)d
From the given,
a1 = 0 ;
n = 16 ;
d = 2
a16 = 0 + (16 – 1)2
a16 = 15 × 2
a16 = 30
Question 2: Find the sum of the first 30 terms of the following sequence.
4, 7, 10, 13,….
Solution:
The given arithmetic sequence is:
4, 7, 10, 13,….
Here,
a1 = 4
d = a2 – a1 = 7 – 4 = 3
n = 30
an = a1 + (n – 1)d
a30 = 4 + (30 – 1)3
= 4 + (29)3
= 4 + 87
= 91
Sum of the first 30 terms
S30Â = n/2[a1 + a30]
= (30/2)[4 + 91]
= 15 × 95
= 1425
Question 3: What is the 25th term of the arithmetic sequence 21, 15, 9, 3, ….?
Solution:
Given the arithmetic sequence is:
21, 15, 9, 3,…
Here, a1 = 21
d = a2Â – a1 = 15 – 21 = -6
nth term
an = a1 + (n – 1)d
25th term of the given sequence is:
a25 = a1 + (25 – 1)d
= 21 + 24(-6)
= 21 – 144
= -123
| More topics in Arithmetic Sequence Formula | |
|---|---|
| Sum of Arithmetic Sequence Formula | Arithmetic Sequence Explicit Formula |
| Arithmetic Sequence Recursive Formula | |
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