Series / Sequences: Determine whether the number is an element in the sequence

Let's check if the number 30 is a term in the sequence defined by the given general term:

$$ $a_n= 15n$,

We will do this in the following way:

We will require first the existence of such a term in the sequence, at some position, meaning we will require that:

$a_n=30$

Then, we will solve the equation obtained from this requirement, while remembering that n is the position of the term in the sequence (also called - the index of the term in the sequence), and therefore must be a natural number, meaning a positive whole number, and therefore we will require this as well,

Let's check if these two requirements are met together:

First, let's solve:

$\begin{cases} a_n= 15n \\ a_n=30 \end{cases}\\ \downarrow\\ 30=15n$where we substituted $a_n$in the first equation with its value from the second equation,

We got an equation with one unknown for n, let's solve it in the regular way by moving terms and isolating the unknown and we get:

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

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

We got therefore that the requirement that:

$a_n=30$

leads to:

$n=2$

and this is indeed a natural number, meaning - positive and whole, and 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.

We know that the first term of the series is 15.

From here we can easily write the entire series, until we see if we reach 1.  

15, 13, 11, 9, 7, 5, 3, 1

The number 1 is indeed an element of the series!

Let's check if the number 15 is a member of the sequence defined by the given general term:

$$ $a_n=n+5$

We will do this in the following way:

We will require first the existence of such a member in the sequence, at some position, meaning we will require that:

$a_n=15$

Then, we will solve the equation obtained from this requirement, while remembering 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, meaning a positive whole number, and therefore we will require this as well,

Let's check if these two requirements are fulfilled together:

First, let's solve:

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

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

We got an equation with one unknown for n, let's solve it in the regular way by moving terms and isolating the unknown and we get:

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

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

We got therefore that the requirement that:

$a_n=15$

leads to:

$n=10$

and this is indeed a natural number, meaning - positive and whole, and therefore we can conclude that in the sequence defined in the problem by the given general term, the number 15 is indeed a member and its position is 10, meaning - in mathematical notation:

$a_{10}=15$

Therefore the correct answer is answer A.