Question

For the sequence ${{a}_{n}}=4n+5$, how do you find the first five terms?

Answer 1

Hint: In the given general form of sequence ${{a}_{n}}=4n+5$ we apply the values to $n.$ $n$ is a natural number. So, the value of $n$ starts from $1.$ Then we give the values $2,3,4$ and $5$ to get the first five terms.

Complete step by step solution:
Consider the given general form of sequence ${{a}_{n}}=4n+5$ for which we are asked to find the first five terms.
In this general form, we apply values, that is natural numbers, to $n.$
When we do this, we get the terms of the given sequence.
What we have to do is to apply \[1,2,3,4\] and $5$ to $n.$
And as a result, we get the terms ${{a}_{1}},{{a}_{2}},{{a}_{3}},{{a}_{4}}$ and ${{a}_{5}}.$
Let us begin with $n=1.$
Then we get the first term ${{a}_{1}}.$
So, the first term is ${{a}_{1}}=4\times 1+5$
And this will give us ${{a}_{1}}=4+5=9.$
Similarly, we can find the second term by applying $2$ to $n.$
Therefore, the second term is ${{a}_{2}}=4\times 2+5$
From this, we will get ${{a}_{2}}=8+5=13.$
We continue this procedure three more times to get the third term, the fourth term, and the fifth term.
Let us apply $3$ to $n.$
Then we will get ${{a}_{3}}=4\times 3+5.$
Now we will get the third term, ${{a}_{3}}=12+5=17.$
We keep doing this until we get the next two terms.
To find the fourth term, we will substitute $n=4$ in the given general term.
And thus, we will get the fourth term as ${{a}_{4}}=4\times 4+5.$
After the calculation, we will get ${{a}_{4}}=16+5=21.$
And finally, the fifth term is ${{a}_{5}}=4\times 5+5.$
That is, ${{a}_{5}}=20+5=25.$
So, we got the first five terms of the sequence ${{a}_{n}}=4n+5.$

Hence, the terms are ${{a}_{1}}=9,{{a}_{2}}=13,{{a}_{3}}=17,{{a}_{4}}=21$ and ${{a}_{5}}=25.$

Note: The sequence given above is an arithmetic progression with the common difference $4.$ The common difference is obtained as ${{a}_{n}}-{{a}_{n-1}}.$ So, we get $25-21=21-17=17-13=13-9=4.$