Verify if 30 is a Term in the Sequence an = 15n: Term Investigation

Determine whether the number 30 is a term in the sequence defined by the given general term:

$$ $a_n= 15n$,

This can be achieved in the following way:

To begin with we require that such a term exists in the sequence, regardless of its position. Hence the expression below.

$a_n=30$

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

Let's check if these two requirements can both be met:

First, let's solve:

$\begin{cases} a_n= 15n \\ a_n=30 \end{cases}\\ \downarrow\\ 30=15n$

When we substituted $a_n$in the first equation with its value from the second equation,

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

$30=15n \\ -15n=-30 \hspace{8pt} \text{/:}(-15)\\ n=2$

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

We thus met the requirement that:

$a_n=30$

Which is turn equals:

$n=2$

This is indeed a natural number - positive as well as whole. Therefore we can conclude that in the sequence defined in the problem by the given general term, the number 30 is indeed a term and its position is 2, meaning - in mathematical notation:

$a_{2}=30$

Therefore the correct answer is answer B.