Arithmetic Sequence Formulas

An arithmetic sequence is a sequence of numbers in which the difference between every two consecutive terms is the same.

Example: 3, 7, 11, 15, 19, ...
Here, the common difference is 4 because each term increases by 4.

The following formulas are commonly used for arithmetic sequences and series:

• a_n is used to find the nth term,
• S_n is used to find the sum of the first n terms,
• d denotes the common difference of an arithmetic sequence.

Example 1: Find the 10^th term of an arithmetic sequence where the first term is 5, and the common difference is 3.

Solution:

The formula for the nth term of an arithmetic sequence is: a_n = a₁ + (n - 1) · d

Here, a₁ = 5, d = 3, and n = 10.
a₁₀ = 5 + (10 - 1) · 3
⇒ a₁₀ = 5 + 9 · 3
⇒ a₁₀ = 5 + 27
⇒ a₁₀ = 32

So, the 10^th term is 32.

Example 2: Find the sum of the first 15 terms of an arithmetic sequence where the first term a₁ is 7 and the common difference d is 4.

Solution:

The formula for the sum of the first n terms Sn of an arithmetic sequence is:

S_n = (n/2) · (2a₁ + (n - 1) · d)

Here, a1 = 7, d = 4, and n = 15.

S₁₅ = (15/2) · (2 · 7 + (15 - 1) · 4)
⇒ S₁₅ = (15/2) · (14 + 14 · 4)
⇒ S₁₅ = (15/2) · (14 + 56)
⇒ S₁₅ = (15/2) · 70
⇒ S₁₅ = 15 · 35
⇒ S₁₅ = 525

So, the sum of the first 15 terms is 525.

Example 3: Find the common difference of an arithmetic sequence where the 3^rd term is 11 and the 7^th term is 23.

Solution:

The formula for the n^th term of an arithmetic sequence is: a_n = a₁ + (n - 1) · d

Let a₃ = 11 and a₇ = 23.

From the formula for the n^th term, we have two equations:

• a₃ = a₁ + 2d
• a₇ = a₁ + 6d

Subtracting the first equation from the second:

a₇ - a₃ = (a₁ + 6d) - (a₁ + 2d)
⇒ 23 - 11 = 4d
⇒ 12 = 4d
⇒ d = 3

So, the common difference is 3.

Apart from these there are other formulas too, which are as:

1. nth term when another term is known

If a_m is known:

\boxed{a_n=a_m+(n-m)d}

2. nth term from the end

If the last term is a_n, then the rth term from the end is:

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

This is particularly useful in problems where the first term isn't given.

3. Sum using the first and last terms

\boxed{S_n=\frac{n}{2}(a_1+a_n)}

4. Number of terms

If a₁ a_n, and d are known:

\boxed{n=\frac{a_n - a_1}{d} + 1}

Practice Questions on Arithmetic Sequence & Series