The general term, given two terms

On this page we calculate the general term of an arithmetic sequence when we know any two terms from the sequence and their positions in the sequence.

In the figure below we are given two terms of an arithmetic sequence, namely u_3 and u_7.

u_3=16. The term value is 16 and n=3 meaning its position is the third term.

u_7=52. The term value is 52 and n=7 meaning its position is the seventh term.

First, let’s find the common difference.

The difference between 52 and 16 is 36.

    \[52-16=36\]

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There are four ‘steps’ between u_3 and u_7. So we divide the difference into four.

    \[36 \div 4 = 9\]

The common difference between consecutive terms is 9.

Calculate the general term of an arithmetic sequence by first finding the common difference

We can find the common difference using a calculation like this:

    \[d=\dfrac{u_7-u_3}{7-3}=\dfrac{52-16}{4}=\dfrac{36}{4}=9\]

Since we don’t know the first term in the sequence, we can either

  • figure it out using mental arithmetic,
  • use an equation.

Let’s use an equation.

Since the common difference is 9, we know that the general term is u_n=9n+b. We don’t yet know the value of b.

To calculate b, let’s substitute one of our known terms: u_3=16. Here, the term is 16 and it is in the third place, so n=3.

    \begin{align*} u_n&=9n+b\\ 16&=9(3)+b\\ 16&=27+b\\ -11&=b\\ \text{Therefore, } u_n&=9n-11 \end{align}

We can check this expression with the other known term. Let’s make sure that this expression gives us 52 for the seventh term.

    \begin{align*} u_n&=9n-11\\ u_7&=9(7)-11\\ &=63-11\\ &=52\quad \checkmark \end{align}

Here’s the summary:

  • calculate the common difference d using any two known terms and their position in the sequence;
  • second, calculate the formula for the general term using one known value and its position in the sequence;
  • lastly check the general term using a different known value and its position in the sequence.

Practice:

Calculate the general term of an arithmetic sequence given any two terms. 

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