Tick the arithmetic sequences.
In order for a sequence to be arithmetic the relations between their numbers must be the sum of a common ratio, therefore to determine which ones are arithmetic sequence we need to satisfy the following expression:
![a_n - a_{n-1} = a_{n-2} - a_{n-3}](https://tex.z-dn.net/?f=a_n%20-%20a_%7Bn-1%7D%20%3D%20a_%7Bn-2%7D%20-%20a_%7Bn-3%7D)
The expression just mean that the subtraction of consecutive terms should be equal for position in the sequence.
"1, 5, 9, 13...":
![13 - 9 = 5 - 1\\4 = 4](https://tex.z-dn.net/?f=13%20-%209%20%3D%205%20-%201%5C%5C4%20%3D%204)
Since the expression is valid, then this is a arithmetic sequence.
"6, 10, 15, 21...:
![21 - 15 = 10 - 6\\6 = 4](https://tex.z-dn.net/?f=21%20-%2015%20%3D%2010%20-%206%5C%5C6%20%3D%204)
Since the expression is invalid, then this isn't a arithmetic sequence.
"2, 3, 5, 8...:
![8 - 5 = 3 - 2\\3 = 1](https://tex.z-dn.net/?f=8%20-%205%20%3D%203%20-%202%5C%5C3%20%3D%201)
Since the expression is invalid, then this isn't a arithmetic sequence.
"2, -4, 8, -16...:
![-16- 8 = -4 - 2\\-24 = -6](https://tex.z-dn.net/?f=-16-%208%20%3D%20-4%20-%202%5C%5C-24%20%3D%20-6)
Since the expression is invalid, then this isn't a arithmetic sequence.
"-1, 2, 5, 8...:
![8 - 5 = 2 -(- 1)\\3 = 3](https://tex.z-dn.net/?f=8%20-%205%20%3D%202%20-%28-%201%29%5C%5C3%20%3D%203)
Since the expression is valid, then this is a arithmetic sequence.
"73, 66, 59, 52...:
![52 - 59 = 66 - 73\\-7 = -7](https://tex.z-dn.net/?f=52%20-%2059%20%3D%2066%20-%2073%5C%5C-7%20%3D%20-7)
Since the expression is valid, then this is a arithmetic sequence.
"6, 1, -4, -9...:
![-9 - (-4) = 1 - 6\\-5 = -5](https://tex.z-dn.net/?f=-9%20-%20%28-4%29%20%3D%201%20-%206%5C%5C-5%20%3D%20-5)
Since the expression is valid, then this is a arithmetic sequence.
"1, 2, 4, 8...:
![8 - 4 = 2 - 1\\4 = 1](https://tex.z-dn.net/?f=8%20-%204%20%3D%202%20-%201%5C%5C4%20%3D%201)
Since the expression is invalid, then this isn't a arithmetic sequence.
Find the first five terms of the patterns with these nth terms.
For
:
![3*1 + 3 = 3 + 3 = 6](https://tex.z-dn.net/?f=3%2A1%20%2B%203%20%3D%203%20%2B%203%20%3D%206)
![6 + 3 = 9](https://tex.z-dn.net/?f=6%20%2B%203%20%3D%209)
![9 + 3 = 12](https://tex.z-dn.net/?f=9%20%2B%203%20%3D%2012)
![12 + 3 = 15](https://tex.z-dn.net/?f=12%20%2B%203%20%3D%2015)
![15 + 3 = 18](https://tex.z-dn.net/?f=15%20%2B%203%20%3D%2018)
(6, 9, 12, 15, 18)
For
:
![9*1 - 7 = 9 - 7 = 2](https://tex.z-dn.net/?f=9%2A1%20-%207%20%3D%209%20-%207%20%3D%202)
![2 + 9 = 11](https://tex.z-dn.net/?f=2%20%2B%209%20%3D%2011)
![11 + 9 = 20](https://tex.z-dn.net/?f=11%20%2B%209%20%3D%2020)
![20 + 9 = 29](https://tex.z-dn.net/?f=20%20%2B%209%20%3D%2029)
![29 + 9 = 38](https://tex.z-dn.net/?f=29%20%2B%209%20%3D%2038)
(2,11,20,29,38)
Write down a formula for the nth term of these patterns. The first term is n = 1.
The nth term of any arithmetic sequence is: ![a(n) = a(1)*(n - 1)*r](https://tex.z-dn.net/?f=a%28n%29%20%3D%20a%281%29%2A%28n%20-%201%29%2Ar)
Therefore we need to identify r in each sequence.
For (9,15,21,27,33):
![r = 15 - 9 = 6](https://tex.z-dn.net/?f=r%20%3D%2015%20-%209%20%3D%206)
![a(n) = 1*(n - 1)*6 \\a(n) = 6*n - 6](https://tex.z-dn.net/?f=a%28n%29%20%3D%201%2A%28n%20-%201%29%2A6%20%5C%5Ca%28n%29%20%3D%206%2An%20-%206)
For (-2,-8,-14,-20,-26):
![r = -8 - (-2) = -6](https://tex.z-dn.net/?f=r%20%3D%20-8%20-%20%28-2%29%20%3D%20-6)
![a(n) = 1*(n - 1)*(-6)\\a(n) = -6*n + 6](https://tex.z-dn.net/?f=a%28n%29%20%3D%201%2A%28n%20-%201%29%2A%28-6%29%5C%5Ca%28n%29%20%3D%20-6%2An%20%2B%206)