What is the formula for the arithmetic sequence if the sum of the same sequence is given by ?

The arithmetic sequence formula is used for the calculation of the nth term of an arithmetic progression. The arithmetic sequence is the sequence where the common difference remains constant between any two successive terms. If we want to find any term in the arithmetic sequence then we can use the arithmetic sequence formula. Let us understand the arithmetic sequence formula using solved examples.

What Is the Arithmetic Sequence Formula?

An Arithmetic sequence is of the form: a, a+d, a+2d, a+3d,......up to n terms. The first term is a, the common difference is d, n = number of terms. For the calculation using the arithmetic sequence formulas, identify the AP and find first term, number of terms and the common difference. There are different formulas associated with an arithmetic series used to calculate the nth term, sum, or the common difference of a given arithmetic sequence.

Arithmetic Sequence Formula

The arithmetic sequence formula is given as,

Formula 1: The arithmetic sequence formula is given as,

\(a_{n}=a_{1}+(n-1) d\)

where,

• \(a_{n}\) = nth term,
• \(a_{1}\) = first term, and
• d is the common difference

The above formula is also referred to as the nth term formula of an arithmetic sequence.

Formula 2: The sum of first n terms in an arithmetic sequence is given as,

\(S_{n}=\frac{n}{2}[2 a+(n-1) d]\)

where,

• \(S_{n}\) =  Sum of n terms,
• \(a_{1}\) = first term, and
• d is the common difference between the successive terms

Formula 3: The formula for calculating the common difference of an AP is given as,

\(d=a_{n}-a_{n-1}\)

where,

• \(a_{n}\) = nth term,
• \(a_{n-1}\) = second last term, and
• d is the common difference between the successive terms

Formula 4: The sum of first n terms of an arithmetic progression when the first and the last terms are known is given as,

\(s_{n}=\frac{n}{2}\left[a_{1}+a_{n}\right]\)

where,

• \(S_{n}\) = Sum of first n terms
• \(a_{n}\) = last term, and
• \(a_{1}\) = first term

Applications of Arithmetic Sequence Formula

We use the arithmetic sequence formula every day or even every minute without even realizing it. Given below are a few real-life applications of the arithmetic sequence formula

• Stacking the cups, chairs, bowls, or a house of cards.  
• Seats in a stadium or an auditorium are arranged in Arithmetic sequence.  
• The seconds' hand on the clock moves in Arithmetic Sequence, so do the minutes' hand and the hour hand.
• The weeks in a month follow the AP, so do the years. Each leap year can be determined by adding 4 to the previous leap year. 
• The number of candles blown on a birthday increases with arithmetic sequence every year.

Let us have a look at a few solved examples to understand the arithmetic sequence formula better.

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Examples Using Arithmetic Sequence Formula

Example 1: Using the arithmetic sequence formula, find the 13th term in the sequence 1, 5, 9, 13...

Solution:     

To find: 13th term of the given sequence.

Since the difference between consecutive terms is same, the given sequence forms arithmetic sequence.

a = 1, d = 4

Using arithmetic sequence formula

\(a_{n}=a_{1}+(n-1) d\)

For 13th term, n = 13

\(a_{n}\) = 1 + (13 - 1)4

\(a_{n}\) = 1 + (12)4

\(a_{n}\) = 1 + 48

\(a_{n}\) = 49

Answer: 13th term in the sequence is 49.

Example 2: Find the first term in the arithmetic sequence where the 35th term is 687 and the common difference 14. 

Solution:     

To find: The first term in the arithmetic sequence

Given: \(a_{n}\) = nth term, d = 14

Using Arithmetic Sequence Formula

\(a_{n}\)=\(a_{1}\)+(n−1)d

687 = \(a_{1}\) + (35 - 1)14

687 = \(a_{1}\) + (34)14

687 = \(a_{1}\) + 476

\(a_{1}\) = 211

Answer: The first term in the sequence is 211.

Example 3: Find the sum of the first 25 terms of the following sequence: 3, 7, 11,.......

Solution:   

To find the sum of the first 25 terms of the arithmetic sequence 3, 7, 11,.......

Given: \(a_{1}\) = 3, d = 4, n = 25

The given arithmetic sequence is 3, 7, 11,….

Using the Sum of Arithmetic Sequence Formula

\(S_{n}=\frac{n}{2}[2 a+(n-1) d]\)

The sum of the first 25 terms

\(S_{25}=\frac{25}{2}[2x3+(25-1) 4]\)

= (25/2)[6+24x4]

= 25/2 × 102

= 1275

Arithmetic sequence formula refers to the formula to calculate the general term of an arithmetic sequence and the sum of the n terms of an arithmetic sequence.

What Is n in Arithmetic Sequence Formula?

In the arithmetic sequence formula for finding the general term,\(a_{n}=a_{1}+(n-1) d\), n refers to the number of terms in the given arithmetic sequence.

What Is the Arithmetic Sequence Formula for the Sum of n Terms?

The sum of first n terms in an arithmetic sequence is given as, \(S_{n}=\frac{n}{2}[2 a+(n-1) d]\) where \(S_{n}\) =  Sum of n terms, \(a_{1}\) = first term, and d is the common difference.

How To Use the Arithmetic Sequence Formula?

Identify that the sequence is an AP and then follow the easy steps given below depending on the values known or given:

• The arithmetic sequence formula is given as, \(a_{n}=a_{1}+(n-1) d\) where, \(a_{n}\)= a general term, \(a_{1}\) = first term, and and d is the common difference. This is to find the general term in the sequence.
• The sum of first n terms in an arithmetic sequence is given as, \(S_{n}=\frac{n}{2}[2 a+(n-1) d]\) where, \(S_{n}\) =  Sum of n terms, \(a_{1}\) = first term, and and d is the common difference
• The formula for calculating the common difference of an arithmetic sequence is given as,\(d=a_{n}-a_{n-1}\) where, \(a_{n}\) = nth term, \(a_{n-1}\) = second last term, and d is the common difference
• The sum of first n terms of an arithmetic progression when the nth term, \(a_{n}\)  is known is given as, \(s_{n}=\frac{n}{2}\left[a_{1}+a_{n}\right]\) = Sum of first n terms, \(a_{n}\) = last term, and, \(a_{1}\) = first term.