Question

How do you find the next three terms of the arithmetic sequence 2.5, 5, 7.5, 10, ……?

Answer 1

Hint: The given sequence is said to be an arithmetic sequence because each term is increasing with a common number, 2.5 . This can be found out by subtracting the first term from the second term. Now use the formula for any term in the arithmetic sequence and then substitute the common difference and the other variables to get the next three terms of the sequence.

Complete step-by-step solution:
The given sequence is an arithmetic sequence.
It means that every term is increasing with a common number.
The given sequence is, 2.5, 5, 7.5, 10, …...
The common difference or the number with which each term gets increased is found out by subtracting the first term with the second term or else the second with the third term and so on.
Therefore, the common difference is, $5-2.5=2.5$ or $7.5-5=2.5$
The formula for any term in an arithmetic sequence is given by,
${{a}_{n}}={{a}_{1}}+\left( n-1 \right)d$
Here ${{a}_{n}}$ is the nth term we have to find and ${{a}_{1}}$ is the first term.
And also $n$ is the term’s place in the sequence and $d$ is the common difference.
We are asked to find the next three terms of the sequence.
That means we have to find the 5,6,7th terms.
So, we need to find, ${{a}_{5}},{{a}_{6}},{{a}_{7}}$
So first let us find ${{a}_{5}}$
$\Rightarrow {{a}_{5}}=2.5+\left( 5-1 \right)2.5$
On evaluating we get,
$\Rightarrow {{a}_{5}}=2.5+\left( 4\times 2.5 \right)$
$\Rightarrow {{a}_{5}}=2.5+10=12.5$
Hence the fifth term will be 12.5
Now let us find ${{a}_{6}}$
$\Rightarrow {{a}_{6}}=2.5+\left( 6-1 \right)2.5$
On evaluating we get,
$\Rightarrow {{a}_{5}}=2.5+\left( 5\times 2.5 \right)$
$\Rightarrow {{a}_{5}}=2.5+12.5=15$
Hence the fifth term will be 15.
Now let us find ${{a}_{7}}$
$\Rightarrow {{a}_{7}}=2.5+\left( 7-1 \right)2.5$
On evaluating we get,
$\Rightarrow {{a}_{5}}=2.5+\left( 6\times 2.5 \right)$
$\Rightarrow {{a}_{5}}=2.5+15=17.5$
Hence the fifth term will be 17.5
Therefore, the next three terms of the sequence are, 12.5, 15, 17.5

Note: The next three terms can be easily found by just adding 2.5 to the last digit and so on. We use this formula for finding the nth term in an arithmetic sequence because sometimes if we have to find the value of the $167^{th}$ term in a sequence, it is difficult to keep on adding 2.5 . Hence, we use this formula to easily evaluate.