Verified answer

Answer:

Step-by-step explanation:

I will not help you to solve the entire 3 terms but I will help you to solve the remaining for your learning process

1. 1,4,7,10,__, __, __

Logic: For problem 1 you can see that there is a pattern, when you subtract 4 and 1, it will give 3, same as 7-4 and 10-7, it is trivial that the difference is 3, what you need to do is add 3 to the last term which is 10. Do you get it?

10+3= 13

Then find the next 2 Terms

Mathematical: We all determine that It is Arithmetic Sequence

The Formula for Arithmethic Sequence is a(n)=a1+(n-1)d.

a1= First Term

d= Difference

Substitute the Values

a(n)=1+(n-1)3

Distribute the Value (n-1)3 = 3*n and 3*-1

a(n)=1+3n-3

a(n)=1+3n-3

The Formula is a(n)= 3n-2

TO VERIFY;

We will find the First term which is a1, which is obviously 1

a(n)= 3n-2

Substitute

a(1)= 3(1)-2

a(1)= 3-2

a1=1; Correct 1,4,7,10,__, __, __

The remaining is

a(5)= 3(5)-2

a(6)= 3(6)-2

a(7)= 3(6)-2

2. 4,-1, -6, -11, __, __, __,

For this item you need to add -5 to each term because -1-4 = 5 as same as -6-(-1) and -11-(-6) therefore

-11+-5= -16

Your Turn!

Mathematical: We all determine that It is Arithmetic Sequence

The Formula for Arithmethic Sequence is a(n)=a1+(n-1)d.

a1= First Term

d= Difference

Substitute the Values

a(n)=4+(n-1)-5

Distribute the Value (n-1)-5 = -5*n and -5*-1

a(n)=4-5n+5

Formula: a(n)=-5n+9

You may solve the 5, 6 and 7th term:

a(5)=-5(5)+9

a(6)=-5(6)+9

a(7)=-5(7)+9

3. 2,6,10,14,___, ___, __,

For this item, you need to add 4 to each term

14+4=18

The Formula for Arithmethic Sequence is a(n)=a1+(n-1)d.

a1= First Term = 2

d= Difference = 4

Substitute the Values

a(n)=2+(n-1)4

Distribute the Value (n-1)4 = 4*n and 4*-1

a(n)=2+4n-4

Formula: a(n)=4n-2

Remaining

a(5)=4(5)-2

a(6)=4(7)-2

a(7)=4(7)-2

Your Turn!

4. 5,-10, 20,-40, __, _, _

For this item, you need to multiply it by -2 because -10/5 = 2 same as the others. This is a geometric sequence.

-40 x  -2 = 80

Formula: [tex]a_n=a_1r^{n-1}\\[/tex]

The ratio is - 2

The first term is 5

Therefore;

[tex]a_n=5\times-2^{n-1}[/tex] is the Formula

Your Turn!

[tex]a_5=5\times-2^{5-1}\\a_6=5\times-2^{6-1}\\a_6=5\times-2^{7-1}[/tex]

5. 1,2,4,8,___, __, __

For this item, you need to multiply it by 2 because 2/1 = 2 same as the others. This is also a geometric sequence.

8 x 2 =16

Formula: [tex]a_n=a_1r^{n-1}\\[/tex]

The ratio is 2

The first term is 1

Therefore;

 Formula: [tex]a_n=1\times2^{n-1}\\[/tex]

Solve the Next:

[tex]a_5=1\times2^{5-1}\\a_6=1\times2^{6-1}\\a_7=1\times2^{7-1}\\[/tex]

You may use the solution in the image.