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

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!

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.

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.