Question

Write an equation for the nth term of each arithmetic sequence.

$28,25,22,19,\mathrm{...}$

Short answer

The equation for nth term of the given arithmetic sequence is $-3n+31$.

Step-by-step solution

Step 1. Find the common difference (d) of the given arithmetic sequence.

The given arithmetic sequence is:$28,25,22,19,\mathrm{...}$

Therefore, it can be written that:

$$\begin{gathered} a_1=28 \\ {a}_2=25 \\ {a}_3=22 \\ {a}_4=19 \end{gathered}$$

The common difference (d) of the arithmetic sequence is the difference between any term and the term prior to that term.

Therefore,

$$\begin{gathered} d=a_2-a_1 \\ =25-28 \\ =-3 \end{gathered}$$

Therefore, the common difference (d) of the given arithmetic sequence is $-3$.

Step 2. Write the equation for the nth term of the arithmetic sequence.

The nth term of the arithmetic sequence $\left(a_n\right)$is given by:

$a_n=a_1+\left(n-1\right)d$

Where,$a_1$ is the first term of the arithmetic sequence and d is the common difference of the arithmetic sequence.

Step 3. Write an equation for the nth term of the given arithmetic sequence having a1=28 and d=-3.

It is given that$a_1=28$ and $d=-3$.

The equation for nth term of the given arithmetic sequence is given by:

$$\begin{gathered} a_n=a_1+\left(n-1\right)d \\ =28+\left(n-1\right)\left(-3\right) \\ =28-3n+3 \\ =-3n+28+3 \\ =-3n+31 \end{gathered}$$

Therefore, the equation for nth term of the given arithmetic sequence is $-3n+31$.