Algebra 2/Trigonometry
Unit 2
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Arithmetic Sequences
Recursive Equations
We know the explicit equation for an arithmetic sequence is:
t(n) = a + d∙n
where a = initial value
d = common difference
We can also write equations for arithmetic sequences another way. A recursive equation is
written as a combination of two equations:
t(0) = a
t(n) = t(n-1) + d
where a = initial value
d = common difference
A recursive equation tells the first term (or initial value), then it gives the rule for finding each
subsequent value. t(n) = t(n-1) + d literally says, “to find the nth term, use the term before it and
add the common difference.” For example, to find the 8th term, use the 7th term and add the
common difference.
Ex:
Make a table for the first 5 terms for the sequence:
t(0) = 5
answer:
n
t(n) = t(n-1) + 4
0
1
2
3
4
t(n)
5
9
13
17
21
Ex:
Write a recursive equation for the following sequence:
4, 1, -2, -5, -8, -11
answer:
t(0) = 4
t(n) = t(n-1) + -3
Ex:
Rewrite the explicit equation as a recursive equation:
t(n) = -6 + 8n
answer:
t(0) = -6
t(n) = t(n-1) + 8
Ex:
Use the last example and find the first 10 terms:
answer:
-6, 2, 10, 18, 26, 34, 42, 50, 58, 66
Given the recursive equation, find the first 5 terms of the arithmetic sequence.
1. t(0) = 0.2
2. t(0) = -100
3. t(0) = 80
t(n) = t(n-1) + 0.4
t(n) = t(n-1) + 7
t(n) = t(n-1) – 12
Given the explicit equation, find the first 5 terms of the arithmetic sequence. Then, rewrite the
equation recursively.
4. t(n) = 4 + 3n
5. t(n) = -2 + 5n
6. t(n) = -18 – 6n
7. t(n) = 27 + 19n
8. t(n) =
7 1
 n
4 4
9. t(n) = 1000 + -
1
n
2
Given the first 5 terms of the arithmetic sequence, write the equation explicitly and recursively.
10. 4, -2, -8, -14, -20, …
11. 5, 6, 7, 8, 9, …
12. 30, 21, 12, 3, -6, …
13. 23, 123, 223, 323, 423, …
14. -8, -11, -14, -17, -20,…
15.
16. -0.5, -0.1, 0.3, 0.7, 1.1, …
17. 1, 6, 11, 16, 21, …
18. 22, -15, -52, -89, -126
1 5 7
11
, , ,1.5, , …
2 6 6
6
Find the explicit equation for the arithmetic sequence that contains the given terms. (refer to
BB41, if necessary). Then, rewrite it as a recursive equation.
19. t(4) = -3
t(62) = -32
20. t(10) = 5
t(28) = 41
21. t(7) = 22
t(37) = -98