# Write the first five terms of the arithmetic sequence. Find the common difference and write the...

## Question:

Write the first five terms of the arithmetic sequence. Find the common difference and write the {eq}n {/eq}th term of the sequence as a function of {eq}n {/eq}.

{eq}a_1 = 15, a_{k+1} = a_k +4 {/eq}

## Arithmetic Sequence:

An arithmetic sequence is a sequence that is created by consecutively adding the same value to the previous number in the sequence. We can write this recursively as

{eq}\begin{align*} a_{n+1} &= a_n +d \end{align*} {/eq}

where {eq}d {/eq} is the common difference. It is often useful to get a formula for the nth term in a sequence that is explicit rather than recursive. We can always write an arithmetic sequence as

{eq}\begin{align*} a_n &= a_1 + (n-1)d \end{align*} {/eq}

Oddly, they start off using n as the dummy variable and then abruptly switch it to k. This is needlessly confusing, so let's write everything in terms of n.

{eq}\begin{align*} a_1 &= 15 \\ a_{n+1} &= a_n +4 \end{align*} {/eq}

Now, we can read straight from the recursive formula that our common difference is clearly {eq}d=4 {/eq}. The first five terms of our sequence are

{eq}\begin{align*} a_1 &= 15 \\ a_2 = 15+4 &= 19 \\ a_3 = 19 + 4 &= 23 \\ a_4 = 23+4 &= 27 \\ a_5 = 27+4 &= 31 \end{align*} {/eq}

We use {eq}d=4 {/eq} and the first term in the sequence to write the general formula for the nth term in our sequence as

{eq}\begin{align*} a_n &= a_1 + (n-1)d \\ &= 15 + (n-1)(4) \\ &= 15 +4n-4 \\ &= \boldsymbol{ 11+4n } \end{align*} {/eq}