Series / Sequences - Examples, Exercises and Solutions

It is possible to see that there is a difference of one number between each number.

That is, 1 is added to each number and it will be the next number:

$1+1=2$

$2+1=3$

$3+1=4$

Etcetera. Therefore, the next numbers missing in the sequence will be:$8+1=9$

$10+1=11$

One can observe that the difference between each number is 2.

That is, with each leap the next number increases by 2:

$94+2=96$

$96+2=98$

$98+2=100$

and so forth......

It is possible to see that if you multiply each number from the right column by 2, you get the number from the left column.

That is:$1\times2=2$

$2\times2=4$

$3\times2=6$

Therefore, the number of balls is 2 times greater than the number of courts.

We start with the right column in the exercises.

Between each number there is a jump of +3:$4+3=7$

$7+3=10$

Etcetera.

Now we move to the left column of the exercises.

Between each number there is a jump of +1:

$2+1=3$

$3+1=4$

Now we can figure out which exercise is missing:

The left digit will be:$2-1=1$

The right digit will be:$4-3=1$

And the missing exercise is:$1+1$

It is apparent, that for each successive number, a square is added in length and one in width.

Hence, the rule using the variable n is:

$a(n)=n^2$

Therefore, the eighth term will be:

$n^2=8\times8=16$

We calculate by replacing$n=11$

$2\times11+2=$

First we solve the multiplication exercise and then we add 2:

$22+2=24$

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!

The third term in the sequence is the term $a_3$:

$a_n= \frac{n}{2}$

We need to substitute in the position of the term in the sequence:

$n=3$

Now, using our values:

$a_{\underline{n}}= \frac{\underline{n}}{2} \\ n=\underline{3}\\ \downarrow\\ a_{\underline{3}}=\frac{\underline{3}}{2}$

Now we substitute the position of the term in the sequence (3) in place of n. The substitution is shown with an underline in the expression above.

Therefore, the correct answer is answer C.

The fourth and fifth terms in the sequence are the terms: $a_4,\hspace{4pt}a_5$meaning in the general term formula given:

$a_n=10n-9$we need to substitute the position (of the requested term in the sequence):

$n=4$$$for - $a_4$and-

$$$n=5$for-

$a_5$ Let's do this for the fourth term:

$a_{\underline{n}}= 10\underline{n}-9 \\ n=\underline{4}\\ \downarrow\\ a_{\underline{4}}= 10\cdot\underline{4}-9=40-9\\ a_4=31$when we substituted in place of n the position (of the requested term in the sequence): 4, substitution is shown with an underline in the expression above,

Similarly, for the fifth term, $a_5$we get:

$a_{\underline{5}}= 10\cdot\underline{5}-9=50-9\\ a_5=41$ which means that:

$a_4=31,\hspace{4pt}a_5=41$Therefore the correct answer is answer A.

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.

In order to determine the first five terms in the sequence simply insert their positions into the given formula as shown below:

$a_n=3n+1$

We want to calculate the values of the terms:

$a_1,\hspace{4pt}a_2,\hspace{4pt}a_3,\hspace{4pt}a_4,\hspace{4pt}a_5$

Let's start with the first term in the sequence,

$a_n=3n+1$

We need to insert the position of whichever term that we want to find.

In this case we want to find the first term so we'll substitute as shown below:

$n=1$

Proceed to calculate:

$a_{\underline{n}}= 3\underline{n}+1 \\ n=\underline{1}\\ \downarrow\\ a_{\underline{1}}=3\cdot\underline{1}+1=4$

When we substituted the position in question in the place of n : the substitution is shown with an underline (as shown above),

Repeat this exact action for all the requested terms in the sequence, meaning for the second through fifth terms:

$a_{\underline{2}}=3\cdot\underline{2}+1=7 \\ a_{\underline{3}}=3\cdot\underline{3}+1=10 \\ a_{\underline{4}}=3\cdot\underline{4}+1=13 \\ a_{\underline{5}}=3\cdot\underline{5}+1=16 \\$For the second term $a_2$ we substituted:$n=2$ in to the formula:

$a_n=3n+1$

For the third term $a_3$ we again substituted:$n=3$ and so on for the rest of the requested terms,

To summarize, we determined that the first five terms:

$a_1,\hspace{4pt}a_2,\hspace{4pt}a_3,\hspace{4pt}a_4,\hspace{4pt}a_5$

in the given sequence, are:

$4,\hspace{4pt}7,\hspace{4pt}10,\hspace{4pt}13,\hspace{4pt}16$

Therefore, the correct answer is answer A.