Q18. - Algebra 1

Question

Find the next three terms of each arithmetic sequence.

$-5,-2,1,4,\mathrm{..}$

Step-by-Step Solution

Answer

The next three terms of the given arithmetic sequence are 7, 10 and 13.

Step 1. Find the first term of the given arithmetic sequence.

The given arithmetic sequence is: $-5,-2,1,4,\mathrm{...}$

From the given arithmetic sequence, it can be noticed that the first, second, third, and fourth terms of the arithmetic sequence are $-5,-2,1$ and 4 respectively.

Step2. Find the common difference.

The common difference of the arithmetic sequence is the difference between the succeeding term and its preceding term.

Therefore, the common difference $\left(d\right)$ of the given arithmetic sequence is:

$$\begin{gathered} d=-2-\left(-5\right) \\ =-2+5 \\ =3 \end{gathered}$$

Therefore, the first term and the common difference of the given arithmetic sequence are $-5$ and 3 respectively.

Step 3. Find the next three terms of the given arithmetic sequence.

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

$a_n=a_1+\left(n-1\right)d$, where $a_1$ is the first term and $d$ is a common difference.

The next three terms of the given arithmetic sequence are fifth, sixth and seventh terms.

Therefore, the next three terms of the given arithmetic sequence having $a_1=-5$ and $d=3$ are:

$a_5=a_1+\left(5-1\right)d$

    $$\begin{gathered} =-5+\left(4\right)\left(3\right) \\ =-5+12 \\ =7 \end{gathered}$$

$a_6=a_1+\left(6-1\right)d$

    $$\begin{gathered} =-5+\left(5\right)\left(3\right) \\ =-5+15 \\ =10 \end{gathered}$$

$a_7=a_1+\left(7-1\right)d$

    $$\begin{gathered} =-5+\left(6\right)\left(3\right) \\ =-5+18 \\ =13 \end{gathered}$$

Therefore, the next three terms of the given arithmetic sequence are 7, 10 and 13.