The general term, given two terms

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

First, let’s find the common difference.

The difference between 52 and 16 is 36.

    \[52-16=36\]

.

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.

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
  • use a different technique.

Let’s use a different technique.

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}


Practice

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