Question

Find the next three terms of each arithmetic sequence.

$3,5,7,9,\mathrm{...}$

Short answer

The next three terms of the given arithmetic sequence are 11, 13 and 15.

Step-by-step solution

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

The given arithmetic sequence is:$3,5,7,9,\mathrm{...}$

From the given arithmetic sequence, it can be noticed that the first, second, third, and fourth terms of the arithmetic sequence are 3, 5, 7, and 9 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:

$$\begin{gathered} d=5-3 \\ =2 \end{gathered}$$

Therefore, the first term and the common difference of the given arithmetic sequence are 3 and 2 respectively.

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

The $nth$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 $a_1=3$and $d=2$are:

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

$$\begin{gathered} =3+\left(4\right)\left(2\right) \\ =3+8 \\ =11 \end{gathered}$$

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

$$\begin{gathered} =3+\left(5\right)\left(2\right) \\ =3+10 \\ =13 \end{gathered}$$

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

$$\begin{gathered} =3+\left(6\right)\left(2\right) \\ =3+12 \\ =15 \end{gathered}$$

Therefore, the next three terms of the given arithmetic sequence are 11, 13 and 15.