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:
[tex]a_n - a_{n-1} = a_{n-2} - a_{n-3}[/tex]
The expression just mean that the subtraction of consecutive terms should be equal for position in the sequence.
"1, 5, 9, 13...":
[tex]13 - 9 = 5 - 1\\4 = 4[/tex]
Since the expression is valid, then this is a arithmetic sequence.
"6, 10, 15, 21...:
[tex]21 - 15 = 10 - 6\\6 = 4[/tex]
Since the expression is invalid, then this isn't a arithmetic sequence.
"2, 3, 5, 8...:
[tex]8 - 5 = 3 - 2\\3 = 1[/tex]
Since the expression is invalid, then this isn't a arithmetic sequence.
"2, -4, 8, -16...:
[tex]-16- 8 = -4 - 2\\-24 = -6[/tex]
Since the expression is invalid, then this isn't a arithmetic sequence.
"-1, 2, 5, 8...:
[tex]8 - 5 = 2 -(- 1)\\3 = 3[/tex]
Since the expression is valid, then this is a arithmetic sequence.
"73, 66, 59, 52...:
[tex]52 - 59 = 66 - 73\\-7 = -7[/tex]
Since the expression is valid, then this is a arithmetic sequence.
"6, 1, -4, -9...:
[tex]-9 - (-4) = 1 - 6\\-5 = -5[/tex]
Since the expression is valid, then this is a arithmetic sequence.
"1, 2, 4, 8...:
[tex]8 - 4 = 2 - 1\\4 = 1[/tex]
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 [tex]3n + 3[/tex]:
[tex]3*1 + 3 = 3 + 3 = 6[/tex]
[tex]6 + 3 = 9[/tex]
[tex]9 + 3 = 12[/tex]
[tex]12 + 3 = 15[/tex]
[tex]15 + 3 = 18[/tex]
(6, 9, 12, 15, 18)
For [tex]9n - 7[/tex]:
[tex]9*1 - 7 = 9 - 7 = 2[/tex]
[tex]2 + 9 = 11[/tex]
[tex]11 + 9 = 20[/tex]
[tex]20 + 9 = 29[/tex]
[tex]29 + 9 = 38[/tex]
(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: [tex]a(n) = a(1)*(n - 1)*r[/tex]
Therefore we need to identify r in each sequence.
For (9,15,21,27,33):
[tex]r = 15 - 9 = 6[/tex]
[tex]a(n) = 1*(n - 1)*6 \\a(n) = 6*n - 6[/tex]
For (-2,-8,-14,-20,-26):
[tex]r = -8 - (-2) = -6[/tex]
[tex]a(n) = 1*(n - 1)*(-6)\\a(n) = -6*n + 6[/tex]
