Arithmetic Sequence Formula

The arithmetic sequence formula is used for the calculation of the n^th term and sum 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/the sum of terms 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 = the number of terms. For the calculation using the arithmetic sequence formulas, first identify the first term, the number of terms and the common difference of the sequence. There are different formulas associated with an arithmetic series used to calculate the n^th term, sum, or the common difference of a given arithmetic sequence.

• n^th term is, a_n = a₁ + (n - 1) d
• Sum of n terms is, S_n = (n/2) [2a₁ + (n - 1) d] (or) (n/2) [a₁ + a_n]
• Common difference, d = a_n - a_{n - 1}

In these formulas, a₁ = the first term, d = common difference, and n = number of terms.

Arithmetic Sequence Formula

The arithmetic sequence formulas are given as,

Formula 1: The arithmetic sequence formula to find the n^th term is given as,

a_n = a₁ + (n - 1) d

where,

• a_n = n^th term,
• a₁ = first term, and
• d is the common difference

Formula 2: The sum of first n terms in an arithmetic sequence is calculated by using one of the following formulas:

• S_n = (n/2) [2a₁ + (n - 1) d] (when we know the first term and the common difference)
• S_n=(n/2) [a₁ + a_n] (when the first and the last terms)

where,

• S_n = Sum of n terms,
• a₁ = first term,
• a_n = n^th term, and
• d is the common difference between the successive terms

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

d = a_n - a_{n - 1}

where,

• a_n = n^th term,
• a_{n - 1} = (n - 1)^th term, and
• d is the common difference between the successive terms

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 an arithmetic sequence.
• The seconds' hand on the clock moves in arithmetic Sequence, and so do the minutes' hand and the hour's hand.
• The weeks in a month follow the arithmetic sequence, and 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 as an arithmetic sequence every year.

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Examples Using Arithmetic Sequence Formula

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

Solution:

To find: 13^th term of the given sequence.

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

a = 1, d = 4

Using arithmetic sequence formula,

a_n = a₁+ (n - 1) d

For 13^th term, n = 13

a_n = 1 + (13 - 1)4

a_n = 1 + (12)4

a_n = 1 + 48

a_n = 49

☛ Also Check: Arithmetic Sequence Calculator

Answer: 13^th term in the sequence is 49.

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

Solution:

To find: The first term in the arithmetic sequence

Given: a_n = n^th term, d = 14

Using arithmetic sequence formula,

a_n = a₁ + (n − 1)d

687 = a₁ + (35 - 1)14

687 = a₁ + (34)14

687 = a₁ + 476

a₁ = 211

Answer: The first term in the sequence is 211.

Example 3: Find the sum of the following arithmetic series: 3 + 7 + 11 + ....... (up to 25 terms).

Solution:

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

Given: a₁ = 3, d = 4, n = 25

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

Using the arithmetic series formula:

S_n = (n/2) [2a + (n - 1) d]

The sum of the first 25 terms

S₂₅=(25/2) [2 x 3 + (25 - 1) 4]

= (25/2) [6 + 24 x 4]

= 25/2 × 102

= 1275

Answer: The sum of the given arithmetic series is 1275.