Arithmetic Sequence Formula

The difference between each succeeding term in an arithmetic series is always the same. In other words, an arithmetic progression or series is one in which each term is formed or generated by adding or subtracting a common number from the term or value before it. The nth term of an arithmetic sequence is calculated using the arithmetic sequence formula. A series is the sum of the terms in a sequence.

For example, the sequence 2, 7, 12, 17, 22, 27,..... is an arithmetic sequence because the common difference between each term is 5.

Similarly, the sequence 3, 7, 10, 14, 17, 25, 28 is not an arithmetic sequence because the common difference between each is not a constant.

Arithmetic Formula to Find nth Term

The arithmetic formula to find the nth term of the sequence is as follows:

If the arithmetic sequence is a1, a2, a3, ………..an, whose common difference is d. Then the nth term an is given by the arithmetic sequence formula as follows:

an = a1 + (n - 1) d

Where an is the nth term of an arithmetic sequence.

a1 is the first term of the arithmetic sequence.

n is the number of terms in the arithmetic sequence.

d is the common difference between each term in the arithmetic sequence. 

In general, the nth term of an arithmetic sequence is given as follows:

an = am + (n - m) d

Arithmetic Formula to Find the Sum of n Terms

An arithmetic series is the sum of the members of a finite arithmetic progression.

For example the sum of the arithmetic sequence 2, 5, 8, 11, 14 will be 2 + 5 + 8 + 11 + 14 = 40

Finding the sum of an arithmetic sequence is easy when the number of terms is less. But when we are dealing with a bigger arithmetic sequence where the number of terms is more, then we will use the arithmetic formula to find the sum of n terms.

The arithmetic sequence formula to find the sum of n terms is given as follows:

\[S_{n}=\frac{n}{2}(a_{1}+a_{n})\]

Where Sn is the sum of n terms of an arithmetic sequence.

n is the number of terms in the arithmetic sequence.

a1 is the first term of the arithmetic sequence.

an is the nth term of an arithmetic sequence.

Exercise Problems on Arithmetic Sequence Formula

1. Find the 18th Term of the Given Sequence: 4, 8, 12, 16, 20, ……..

Ans: First check whether the given series is an arithmetic sequence and then proceed to find the required answer. Here the common difference between each term is constant that is 

8 - 4 = 4, 12 - 8 = 4, 16 - 12 =4, ……

Therefore the given series is an arithmetic sequence.

Here the given arithmetic sequence is 4, 8, 12, 16, 20, ……..

From this sequence, the first term is a1 = 4.

The common difference between each term is d = 8 - 4 = 4.

We have to find the 18th term so n = 18.

Now substitute these values into a formula to find nth term.

an = a1 + (n - 1) d

a18 = 4 + (18 - 1) 4

a18 = 4 + (17) 4

a18 = 4 + 68

a18 = 72.

Therefore the 18th term of the given arithmetic sequence is 72.

2. Find the 25th and 40th term of the sequence: 3, 8, 13, 18, 23,........

Ans: Here the common difference between each term is 

8 - 3 =5, 13 - 8 = 5, 18 - 13 = 5, ……

Therefore the given series is an arithmetic sequence.

Here the given arithmetic sequence is 3, 8, 13, 18, 23,........

From this sequence, the first term is a1 = 3.

The common difference between each term is d = 8 - 3 = 5.

We have to find the 25th and 40th term so n = 25 and n = 40.

25th term

The nth term of an arithmetic sequence given by the formula

an = a1 + (n - 1) d

Substituting the values we get

a25 = 3 + (25 - 1) 5

a25 = 3 + (24) 5

a25 = 3 + 120

a25 = 123.

40th term

The nth term of an arithmetic sequence given by the formula

an = a1 + (n - 1) d

Substituting the values we get

a40 = 3 + (40 - 1) 5

a40 = 3 + (39) 5

a40 = 3 + 195

a40 = 198.

Therefore the 25th and 40th term of the arithmetic sequence is 123 and 198 respectively.

3. Find the Sum of the Given Arithmetic Sequence: 1, 8, 15, 22, 29, 36, 43, 50.

Ans: Here the common difference between each term is 

8 - 1 = 7, 15 - 8 = 7, 22 - 15 = 7, 29 - 22 = 7, 36 - 29 = 7, 43 - 36 = 7, 50 - 43 =7.

Since the common difference is constant, therefore the given sequence is an arithmetic sequence.

Here the first term is a1 = 1 and nth term is a8 = 50.

Number of terms is n = 8.

Now substituting the values into the sum of the nth term formula we get

\[S_{n}=\frac{n}{2}(a_{1}+a_{n})\]

\[S_{8}=\frac{8}{2}({1}+{50})\]

S8 = 4(51)

S8 = 204.

Therefore the sum of the arithmetic sequence 1, 8, 15, 22, 29, 36, 43, 50 is 204.

Conclusion

Arithmetic sequences are numbers that are made up of the previous number plus a constant. The most prevalent distinction is the numerical difference. An arithmetic series is formed when a few or all of the numbers in a sequence are added together. In real life, the arithmetic sequence is crucial because it allows us to understand things through patterns. Sequences and series play a significant role in our lives in a variety of ways. They aid in decision-making by assisting us in predicting, evaluating, and monitoring the outcome of a situation or occurrence.