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.

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 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:$1+1$

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.

The third term in the sequence is the term $a_3$meaning in the general term formula given:

$a_n= \frac{n}{2}$We need to substitute the position (of the requested term in the sequence):

$n=3$Let's do this:

$a_{\underline{n}}= \frac{\underline{n}}{2} \\ n=\underline{3}\\ \downarrow\\ a_{\underline{3}}=\frac{\underline{3}}{2}$When we substituted in place of n the position (of the requested term in the sequence): 3, the substitution is shown with an underline in the expression above,

Therefore, the correct answer is answer C.

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 find the first five terms in the sequence by substituting their positions in the given general term formula:

$a_n=3n+1$

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,

meaning in the given general term formula:

$a_n=3n+1$

We need to substitute the position (of the requested term in the sequence),

We want to find the first term so we'll substitute:

$n=1$

Let's perform this:

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

When we substituted in place of n the position (of the requested term in the sequence): 1, the substitution is shown with an underline in the expression above,

We'll repeat this action identically for all the requested terms in the sequence, meaning for the second through fifth terms:

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

$a_n=3n+1$

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

To summarize, we found 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.

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.

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.