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

Examples with solutions for Series / Sequences: Determine whether the number is an element in the sequence

Exercise #1

an=n+5 a_n=n+5

Determine whether the number 15 a term in the sequence above:

Video Solution

Step-by-Step Solution

Determine whether the number 15 is a member of the sequence defined by the following expression:

an=n+5 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:

an=15 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:

{an=n+5an=1515=n+5 \begin{cases} a_n=n+5\\ a_n=15 \end{cases}\\ \downarrow\\ 15=n+5

We inserted an 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+5n=515n=10/:(1)n=10 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:

an=15 a_n=15

Leading to:

n=10 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:

a10=15 a_{10}=15

Therefore the correct answer is answer A.

Answer

Yes

Exercise #2

According to the following rulean=15n a_n= 15n .

Determine whether 30 is a term in the sequence:

Video Solution

Step-by-Step Solution

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

an=15n 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.

an=30 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:

{an=15nan=3030=15n \begin{cases} a_n= 15n \\ a_n=30 \end{cases}\\ \downarrow\\ 30=15n

When we substituted an 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=15n15n=30/:(15)n=2 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:

an=30 a_n=30

Which is turn equals:

n=2 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:

a2=30 a_{2}=30

Therefore the correct answer is answer B.

Answer

Yes, it is the second term.

Exercise #3

Given a series whose first element is 15, each element of the series is less by 2 of its predecessor.

Is the number 1 an element of the series?

Video Solution

Step-by-Step Solution

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!

Answer

Yes

Exercise #4

Look at the sequence below:

... ,1800, 1700, 1600, 1500

Which of the following numbers will appear in the sequence of numbers indicated above?

Video Solution

Answer

2000

Exercise #5

3n3 3n-3

Is the number 0 a term in the sequence above?

Video Solution

Answer

Yes, it's the first term.

Exercise #6

A sequence has a rule of n4 n-4 .

Is the number 10 a term in the sequence?

Video Solution

Answer

Yes, it is the 14th term.

Exercise #7

,,

How many squares are there in the fourth element?

Video Solution

Answer

7

Exercise #8

,

How many triangles are in the third element?

Video Solution

Answer

2

Exercise #9

Given the series whose first element is 10.

Each term of the series is greater by 2.5 of its predecessor.

Is the number 22.5 an element in the series?

If so, please indicate your place in the series.

Video Solution

Answer

Yes, 6 6

Exercise #10

Given a series whose first element is 1.5.

Each element of the series is greater by 3 of its predecessor.

Is the number 29 an element in the series?

If so, please indicate your place in the series.

Video Solution

Answer

No

Exercise #11

The following is a series of structures formed by squares with side lengths of 1 cm.

Is it possible for a structure to have 46 squares? If so, which element of the series is it?

Video Solution

Answer

No

Exercise #12

Here is a series of structures made of squares whose side length is 1 cm.

Is it possible to have a structure in the series that has 49 squares? If so, what element of the series is it?

Video Solution

Answer

Yes, 7 7

Exercise #13

The following is a series of structures formed by squares with side lengths of 1 cm.

Is it possible to have a structure in the series that has 81 squares? If so, what element of the series is it?

Video Solution

Answer

Yes, 9 9

Exercise #14

The following is a series of structures formed by squares with side lengths of 1 cm.

Is it possible to have a structure in the series that has 55 squares? If so, what element of the series is it?

Video Solution

Answer

No

Exercise #15

The following is a series of structures formed by squares with side lengths of 1 cm.

Is it possible to have a structure in the series that has 64 squares? If so, what element of the series is it?

Video Solution

Answer

Yes, 8 8

Exercise #16

Given a formula with a constant property that depends onn n :

2(2n2) 2(2n-2)

Is the number 20 Is it part of the series? If so, what element is it in the series?

Video Solution

Answer

Yes, 6 6

Exercise #17

Given a formula with a constant property that depends onn n :

4n2 4n-2

Is the number 18 Is it part of the series? If so, what element is it in the series?

Video Solution

Answer

Yes, 5 5

Exercise #18

Given a formula with a constant property that depends onn n :

2n+2 2n+2

Is the number 9 Is it part of the series? If so, what element is it in the series?

Video Solution

Answer

No

Exercise #19

Given a formula with a constant property that depends onn n :

n0.5n n-0.5n

Is the number 5 Is it part of the series? If so, what element is it in the series?

Video Solution

Answer

Yes, 10 10

Exercise #20


2n2 2n^2

Is the number 8 a term in the sequence above?

Video Solution

Answer

Yes.