1. sequence

4. 27 22 17 12 7 SS(2) = S(2-1) – 5 = S(1) – 5 =27 – 5 = 22==
= 22 – 5 = 17
S(4) = S(4 – 1) -5
= S(3) – 5
= 17 – 5 =12
S(5) = S(5 – 1) – 5
= S(4) – 5
= 12 – 5 = 7 27 22 17 12 7

5. It is a function because every x coordinate has one y value.
There is also a common difference of 5.
The domain of S : { 1, 2, 3, 4, 5 }

6. Now try this

7. F(n)= 2f(n – 1) + 3n f(1) = -2
f(2) = 2f(2-1) + 3n
= 2f(1) + 3(2)
= 2(-2) + 6
= = 2
F(3) = 2f(3-1) + 3n
= 2(2) + 3(3)
= = 13
F(4) = 2f(4 -1) + 3n
= 2f(3) + 3(4)
= 2(13) + 12
= = 38
F(5) = 2f(5-1) + 3n
=2f(4) + 3(5)
= 2(38) + 15
= = 91
The first 5 terms are { -2, 2, 13, 38, 91}

8. Arithmetic sequence
A sequence is an ordered list of numbers and the sum of the terms of a sequence is a series. In an arithmetic sequence, each term is equal to the previous term, plus (or minus) a constant. The constant is called the common difference (d). 

10. Let’s test your knowledge

11. a3 = a5 = first term = a1
First find the common difference, d, by subtracting the third term from the fifth term.
a5 – a3 = 26 – 10 = 16
However, you must divide by 2 because you were not given the fourth term. There are 2 terms after the third term and since the difference is the same, you must divide by 2.

12. Solve for d: d = 𝟏𝟔 𝟐 = d = 8
Next use either the formula for finding the third of fifth term and the known information to calculate the value of the first term.
a5 = a (n - 1)d
= a1 + (5 – 1)8
26 = a (8)
= a
= a1
-6 = a1
The first term a1 is -6

13. an = a1 + (n – 1)d an = -6 + (n – 1)8 an = -6 + 8n – 8 an = 8n – 8 – 6
Now solve for the equation or finding the nth term
an = a1 + (n – 1)d
an = (n – 1)8
an = n – 8
an = 8n – 8 – 6
an = 8n -14

14. Show your skill

15. a1 = 12, d = 4
an = a1 + (n- 1)d
an = 12 + (n – 1)4
an = n -4
an = 4n + 8

16. Geometric sequence
A geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non- zero number called the common ratio. ... The sum of the terms of a geometric progression, or of an initial segment of a geometric progression, is known as a geometric series.

18. Finding the common ratio
To find the common ratio, r, you must divide 2 consecutive terms. Example: a2 a1

19. Test your skills
This series does not have a common difference, so it is a geometric series. Find the common ratio and substitute it in the formula for a geometric series.

20. This is a geometric series what a common ratio, r, of
-2.
Fomula for a recursive geometric series a(n) = r(an- 1)