1. Explicit formulas
Sequences: Lesson 3

2. Explicit formulas
An Explicit Formula is a formula for calculating each term using the index n (the term number in the sequence)
An explicit formula allows you to find any element of a sequence without knowing the element before it.
Plug in the term number you are trying to find as n, input, and simplify to find an , output. Like plugging an input into a function.
(we have already practiced using explicit formulas!)
Explicit formulas can be used for arithmetic or geometric sequences

3. Use Explicit formulas for arithmetic sequences:
Use the rule an = 2+n
Find a10 (the 10th term)
Plus in 10 as n and evaluate
a10 = 2 + (10) = 12
So the 10th term is 12
a10 = 12
Write the first 5 terms of this arithmetic sequence.
xn = (n-1)
Substitute 1 in for “n.” and evaluate
x1 = (1-1) = (0) = 19
x2 = (2-1) = = 13
and then repeat for n=3 thru 5
19, 13, 7, 1, -5, …

4. Use Explicit formulas for geometric sequences:
Write the first 5 terms of this geometric sequence.
xn = (3)(2)(n-1)
Substitute 1 in for “n.” and evaluate
x1 = (3)(2)(1-1)
x1 = (3)(2)(0)
x1 = (3)(1) = 3
x2 = (3)(2)(2-1) = (3)(2)(1) = 6
and then repeat for n=3 thru 5
Use the rule an = 2n
Find a5 (the 5th term)
Plus in 5 as n and evaluate
a5 = 2(5) = 32
So the 5th term is 32
a5 = 32
3, 6, 12, 24, 48, …

5. Write Explicit formulas for Arithmetic Sequences
Arithmetic sequences work like this: a, a + d, a + 2d, a + 3d, …
a or a1 is the first term.
d is the common difference between terms
To write an arithmetic sequence, the rule is:
an = a1 + d(n-1)
(We use “n-1” because d is not used on the 1st term.)

6. Write Explicit formulas for Arithmetic Sequences
Write the explicit rule for the sequence
19, 13, 7, 1, -5, …
Start with the formula: an = a + d(n-1)
a is the first term = 19
d is the common difference: -6
The rule is:
an = (n-1)

7. Write Explicit formulas for Arithmetic Sequences
Write the explicit rule for the sequence
4, 7, 10, 13, 16
Start with the formula: an = a + d(n-1)
a is the first term = 4
d is the common difference: +3
The rule is:
an = 4 + 3(n-1)
Sequence Term
Term a1 4 a2 7 a3 10 a4 13 a5 16
Notice, this is also the linear function y=3x+1. The 3 is the rate of change but the y-intercept is 1 not 4 because the linear function starts when x=0 and a sequence starts at a1 similar to x=1.

8. Write Explicit formulas for geometric Sequences
Geometric sequences work like this: a, ar, ar2, ar3, …
a (or a1) is the first term
r is the common ratio between terms
To write a geometric sequence, the rule is:
an = a1r(n-1)
(We use “n-1” because r is not used on the 1st term,
so ar0 is the 1st term.)

9. Write Explicit formulas for geometric Sequences
Write the explicit rule for the sequence
3, 6, 12, 24, 48, …
Start with the formula: an = ar(n-1)
a is the first term = 3
r is the common ratio: 2
The rule is:
an = (3)(2)(n-1)
(Order of operations states that we would take care of exponents before you multiply.)

10. Practice 1. Find the 12th term of this sequence. xn = 19 - 6(n-1)
3. Write an explicit formula for the sequence 9, 1, -7, -15…
x12 = -47
x12 = 6,144
an = 9 – 8(n-1)

11. recap