Question

Find the next three terms of each arithmetic sequence.

$1.4,1.2,1.0,\mathrm{...}$

Short answer

The next three terms of the given arithmetic sequence are 0.8, 0.6 and 0.4.

Step-by-step solution

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

The given arithmetic sequence is: $1.4,1.2,\mathrm{1.0...}$

Therefore, it can be written that:

$$\begin{gathered} a_1=1.4 \\ {a}_2=1.2 \\ {a}_3=1.0 \end{gathered}$$

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

Therefore,

$$\begin{gathered} d=a_2-a_1 \\ =1.2-1.4 \\ =-0.2 \end{gathered}$$

Therefore, the common difference (d) of the given arithmetic sequence is $-0.2$.

Step 2. Find the next three terms of the given arithmetic sequence.

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

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

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

The next three terms of the given arithmetic sequence are $a_4,a_5$and $a_6$.

The given arithmetic has first term as 1.4 and common difference as $-0.2$.

Therefore, $a=1.4$and $d=-0.2$.

Therefore, it is obtained that:

$$\begin{gathered} a_4=1.4+\left(4-1\right)\left(-0.2\right) \\ =1.4+3\left(-0.2\right) \\ =1.4-0.6 \\ =0.8 \end{gathered}$$

$$\begin{gathered} a_5=1.4+\left(5-1\right)\left(-0.2\right) \\ =1.4+4\left(-0.2\right) \\ =1.4-0.8 \\ =0.6 \end{gathered}$$

$$\begin{gathered} a_6=1.4+\left(6-1\right)\left(-0.2\right) \\ =1.4+5\left(-0.2\right) \\ =1.4-1.0 \\ =0.4 \end{gathered}$$

Therefore, the next three terms of the given arithmetic sequence are 0.8, 0.6 and 0.4.