1. sequences: Using Explicit and recursive formulas

2. Explicit formulas
An Explicit Formula is a formula for calculating each term using the index (the term number in the sequence)
An explicit formula allows you to find any element of a sequence without knowing the element before it.
Example: Use the rule an = 2n-1 (this is an explicit formula)
Use the rule to find the 10th term of the sequence.
Plus in 10 as n and evaluate
a10 = 2(10-1) = 29 = 256
So the 10th term is 256
a10 = 256

3. Using explicit formulas
xn = (n-1)
Find the 12th term of this sequence.
Substitute 12 in for “n.”
x12 = (12-1)
x12 = (11)
x12 = 19 – 66
x12 = -47
Write the first 5 terms of this arithmetic sequence.
Substitute 1 in for “n.” and evaluate
x1 = (1-1)
x1 = (0)
x1 = 19 – 0
x1 = and then repeat for n=2 thru 5
19, 13, 7, 1, -5, …

4. Using explicit formulas
xn = (3)(2)(n-1)
Find the 12th term of this sequence.
Substitute 12 in for “n.”
x12 = (3)(2)(12-1)
x12 = (3)(2)(11)
x12 = (3)(2048)
x12 = 6,144
Write the first 5 terms of this geometric sequence.
Substitute 1 in for “n.” and evaluate
x1 = (3)(2)(1-1)
x1 = (3)(2)(0)
x1 = (3)(1)
x1 = and then repeat for n=2 thru 5
3, 6, 12, 24, 48, …

5. Recursive formula an = an-1 + 2
A Recursive Formula expresses each term of a sequence based on the preceding term.
A recursive formula always uses the preceding term to define the next term of the sequence.
Recursive formulas look a little different than explicit. They will have an an-1 somewhere in the equation which represents the term prior to the term you are solving for
an = an-1 + 2

6. How does a recursive Formula Work?
an = an-1 + 2
2, 4, 6, 8…
Find the 5th term.
The 4th term in this sequence 8. (a4 = 8)
a5 = a(5-1) + 2
a5 = a4 + 2
a5 = 8 + 2
a5 = 10
an = 2.5(an-1)
a18 = 10
Find the 19th term.
a19 = 2.5(a(19-1))
a19 = 2.5(a18)
a19 = 2.5(10)
a19 = 25
You MUST know the term that came before the one you want

7. Using explicit formulas
an = an-1 – a1 = 27
Find the first 5 terms of the sequence
We are provided the 1st term of the sequence, 27. We need to find the next four terms.
a2 = 27 – 2
a2 = 25
a3 = 25 – 2
a3 = 23
a4 = 23 – 2
a4 = 21
a5 = 21 – 2
a5 = 19
The first five terms of the sequence are 27, 25, 23, 21, and 19.

8. Bouncing Ball
Use the explicit formula that models the height of a bouncing ball
an = 3(0.6)(n-1)
Find the height of the ball after the 4th rebound.
a4 = 3(0.6)(4-1)
a4 = 3(0.6)3
a4 = 3(0.1296)
a4 = feet

9. Option A: a15 = 100(15) + 900 a15 = 1500 + 900 a15 = $2,400 Option B:
Option A: If you deposit $1000, the second day your account will have $1,100, the third day your account will have $1,200, the fourth day your account will have $1,300, and so forth.
an = 100n + 900
Option B: If you deposit $1, the second day your account will have $3, the third day your account will have $9, and the fourth day your account will have $27, and so forth.
an = 1(3)(n-1)
Calculate how much money you will have with each option on the 15th day.
Option A:
a15 = 100(15) + 900
a15 =
a15 = $2,400
Option B:
an = 1(3)(n-1)
a15 = 1(3)(15-1)
a15 = 1(3)(14)
a15 = $4,782,969