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

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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