1. explicit form and vice versa?
Aim: How do we write a recursive sequence in
explicit form and vice versa?
Do Now: Given the sequence formulas
a) 𝑎 𝑛 =2+3(𝑛−1)
b) 𝑎 1 =2
an = an –1 + 3
What is the difference between them?

2. Converting from Recursive to Explicit
Use the following explicit formulas
With a1 as first term (use for 𝑛≥1)
Arithmetic sequence: an = a1 + d(n – 1)
Geometric Sequence: an = a1rn –1
With a0 as first term (use for 𝑛≥0)
Arithmetic sequence: an = a0 + dn
Geometric Sequence: an = a0rn

3. 𝑎) 𝑎 𝑛 =4 𝑎 𝑛−1 with 𝑎 1 =12 𝑎 2 =12∙4 𝑎 3 =12∙4∙4=12∙ 4 2
Convert each of the following recursive formulas to explicit formulas. Identify each sequence as arithmetic, geometric, or neither.
𝑎) 𝑎 𝑛 =4 𝑎 𝑛−1 with 𝑎 1 =12
𝑎 2 =12∙4
𝑎 3 =12∙4∙4=12∙ 4 2
𝒂 𝒏 =𝟏𝟐∙ 𝟒 𝒏−𝟏
𝑏) 𝑎 𝑛 =4.2+ 𝑎 𝑛−1 with 𝑎 1 =12
𝑎 2 =12+4.2
𝑎 3 = =12+4.2(2)
𝒂 𝒏 =𝟏𝟐+𝟒.𝟐(𝒏−𝟏)

4. c) 𝑎 𝑛+1 = 5 𝑎 𝑛 with 𝑎 0 =2
𝑎 1 = 5 ∙2
𝑎 2 = 5 ∙ 5 ∙2=2∙
𝒂 𝒏 =𝟐∙ 𝟓 𝒏
d) 𝑎 𝑛+1 = 𝑎 𝑛 with 𝑎 0 =2
𝑎 1 = 5 +2
𝑎 2 = =2+2 5
𝒂 𝒏 =𝟐+𝒏 𝟓

5. Convert from Explicit to Recursive
Use the following recursive formulas
Arithmetic sequence: an + 1 = an + d
geometric sequence: an + 1 = an r
Write each sequence in recursive form.
𝑎) 𝑎 𝑛 = 𝑛 for 𝑛≥0
𝑎 0 = 1 5
𝑟=3
𝒂 𝒏+𝟏 = 𝒂 𝒏 ∙𝟑
𝑏) 𝑎 𝑛 =16−2𝑛 for 𝑛≥1
𝑎 1 =14
𝑑=−2
𝒂 𝒏+𝟏 = 𝒂 𝒏 −𝟐

6. 𝑐) 𝑎 𝑛 = 𝑛 for 𝑛≥1
𝑟= 1 2
𝑎 1 =
𝑎 1 =8
𝑎 𝑛+1 = 𝑎 𝑛 1 2
𝑑) 𝑎 𝑛 =71− 6 7 𝑛 for 𝑛≥0
𝑑=− 6 7
𝑎 0 =71
𝑎 𝑛+1 = 𝑎 𝑛 − 6 7

7. The first term a0of a geometric sequence is –5, and the common ratio is –2.
a. What are the terms a0, a1, and a2?
𝒂 𝟎 =−𝟓, 𝒂 𝟏 =𝟏𝟎, 𝒂 𝟐 =−𝟐𝟎
b. Find a recursive formula for this sequence.
𝒂 𝟎 =−𝟓
c. Find an explicit formula for this sequence.
𝒂 𝒏+𝟏 = 𝒂 𝒏 (−𝟐)
𝒂 𝒏 = 𝒂 𝟎 −𝟐 𝒏
d. What is term a9?
𝒂 𝟗 =−𝟓 −𝟐 𝟗=−𝟓 −𝟓𝟏𝟐 =𝟐𝟓𝟔𝟎
e. What is term a10?
𝒂 𝟏𝟎 =−𝟓 −𝟐 𝟏𝟎=−𝟓 𝟏𝟎𝟐𝟒 =−𝟓𝟏𝟐𝟎

8. The recursive formula for a geometric sequence is 𝑎 𝑛+1 =3
The recursive formula for a geometric sequence is 𝑎 𝑛+1 =3.92 𝑎 𝑛 with 𝑎 0 = Find an explicit formula for this sequence.
𝒂 𝒏 =𝟒.𝟎𝟓 𝟑.𝟗𝟐 𝒏
The explicit formula for a geometric sequence is 𝑎 𝑛 = 𝑛. Find a recursive formula for this sequence.
𝒂 𝒏 =𝟏𝟒𝟕 𝟐.𝟏 𝟑𝒏
=𝟏𝟒𝟕 𝟔.𝟑 𝒏
𝒂 𝟏 =𝟏𝟒𝟕 𝟔.𝟑 =𝟗𝟐𝟔.𝟏
𝒂 𝟏 =𝟗𝟐𝟔.𝟏
𝒂 𝒏+𝟏 = 𝒂 𝒏 (𝟐.𝟏)