# Write a rule for the {eq}n {/eq}th term of the arithmetic sequence that has terms a{eq}_4 {/eq} = 31 and a{eq}_{10} {/eq} = 85.

## Question:

Write a rule for the {eq}n {/eq}th term of the arithmetic sequence that has terms a{eq}_4 {/eq} = 31 and a{eq}_{10} {/eq} = 85.

## Arithmetic Sequence:

An arithmetic sequence is a sequence that starts at some fixed constant {eq}a_0 {/eq} and adds a constant number, c, each time.

The formula for the {eq}n_{th} {/eq} term is {eq}a_n = a_0 + n*c {/eq}.

The formula for the {eq}n_{th} {/eq} term is {eq}a_n = a_0 + n*c {/eq}.

For n = 4, {eq}a_4 = a_0 + 4*c = 31 {/eq}.

For n = 10, {eq}a_{10} = a_0 + 10*c = 85 {/eq}.

We need to solve these two equations. We can subtract the first from the second and we get:

{eq}6c=54 \\ \Rightarrow c = 9 {/eq}

Now, we plug in the value of c in the first equation to get:

{eq}a_0 + 4*9 = 31 \\ \Rightarrow a_0 = -5 {/eq}

Hence, a formula is:

{eq}a_n = -5 + 9n {/eq}