Sequence Formula an = n + 5: Is 15 a Term in the Series?

$a_n=n+5$

Determine whether the number 15 a term in the sequence above:

Determine whether the number 15 is a member of the sequence defined by the following expression:

$$ $a_n=n+5$

This can be achieved in the following way:

Our first requirement is that the value 15 does in fact exist within the sequence regardless of its position.

Hence the following expression:

$a_n=15$

We will proceed to solve the equation obtained from this requirement. Remember that n is the position of the member in the sequence (also called - the index of the member in the sequence), and therefore must be a natural number ( a positive whole number).

Let's check whether these two requirements can be met:

First, let's solve:

$\begin{cases} a_n=n+5\\ a_n=15 \end{cases}\\ \downarrow\\ 15=n+5$

We inserted $a_n$into the first equation with its value from the second equation.

We obtained an equation with one unknown for n. Let's proceed to solve it by moving terms and isolating the unknown as shown below:

$15=n+5 \\ -n=5-15\\ -n=-10 \hspace{8pt} \text{/:}(-1)\\ n=10$

In the last step we divided both sides of the equation by the coefficient of the unknown on the left side,

Thus we met the requirement that:

$a_n=15$

Leading to:

$n=10$

This is indeed a natural number, - positive and whole. Therefore we can conclude that the number 15 is indeed present in the sequence defined in the problem, and its position is 10, meaning - in mathematical notation:

$a_{10}=15$

Therefore the correct answer is answer A.