# How do you find an equation that describes the sequence 16, 17, 18, 19,... and find the 23rd term?

Jun 30, 2017

${a}_{n} = n + 15$
${a}_{23} = 38$

#### Explanation:

To find an equation, we should use the arithmetic sequence formula:

${a}_{n} = {a}_{1} + \left(n - 1\right) d$

When using this formula, you need to find the values for $d$ and ${a}_{1}$.

1. Finding $d$ (common difference)

We are given the sequence $16 , 17 , 18 , 19 , \ldots$. Using this information, we can find the common difference ($d$), which is another way of saying the difference between any two consecutive numbers in the arithmetic sequence. You could find $d$ in a number of ways:

$17 - 16 = 1$
$18 - 17 = 1$

etc.

But whatever way you choose to find it, you should get that $d = 1$.

2. Finding ${a}_{1}$

${a}_{1}$ is the first term of the sequence. In our case, ${a}_{1}$ = 16.

3. Plug into the formula.

${a}_{n} = 16 + \left(n - 1\right) \left(1\right)$

Now distribute $1$ to $\left(n - 1\right)$:

${a}_{n} = 16 + n - 1$
${a}_{n} = n + 15$

Now plug in 23 for $n$:

${a}_{23} = 23 + 15$
${a}_{23} = 38$

Jun 30, 2017

${a}_{n} = n + 15 \text{ and } {a}_{23} = 38$

#### Explanation:

$\text{ this is an arithmetic sequence}$

$\text{the nth term is } {a}_{n} = a + \left(n - 1\right) d$

$\text{where " a " is the first term and " d" the common difference}$

$\text{here " a=16" and } d = 1$

$\Rightarrow {a}_{n} = 16 + n - 1 = n + 15$

$\Rightarrow {a}_{23} = 23 + 15 = 38$