Arithmetic Sequences

sequence

number pattern

arithmetic sequence

a numerical pattern that adds or subtracts at a constant rate or value

Arithmetic Means

The missing terms in an arithmetic sequence.

term number

The order number in which the term falls in the
sequence (x)

term

Each number in the sequence (y) or f(x)

consecutive terms

a term directly after another term

common difference

the difference between any two consecutive terms in an arithmetic sequence

Find the common difference:
16, 25, 34, ..

9

Find the common difference in the sequence: 10, 3, -4, -11, ...

-7

Find the next three terms in the sequence: 3, 7, 11, 15, ...

19, 23, 27

Find the next three terms in the sequence: 4, -1, -6, ...

-11, -16, -21

What is the Explicit Formula/Function Rule of an arithmetic sequence? And, what does each variable represent?

A(𝑛) = a₁+ d(𝑛-1)
𝑛 = the term number in the sequence (input/x)
A(𝑛)=value of the nth term of the sequence(output/f(x)/y)

Term number: 1 2 3 4 ← input
Term: 7 11 15 19 ←output

Function Rule/Explicit Formula: A(𝑛) = 7 + 4(𝑛-1)
Simplified: A(𝑛) = 4n +3

Input Output
𝑛 A(𝑛)
1 7 7 + 0 (4) 7
2 7 + 4 7 + 1 (4) 11
3 7 + 4 + 4 7 + 2 (4) 15
4 19

Simplify the following arithmetic sequence explicit formulas
1. A(𝑛) = -9 + (𝑛-1)6
2. A(𝑛) = 1/5 + (𝑛 - 1)4/5

1. A(𝑛) = 6𝑛 - 15
2. A(𝑛) = 4/5𝑛 - 3/5

Find the 10th term in the sequence: -5, -2, 1, 4, ...

22

Find the nth term in the sequence. (Write the explicit formula): 10, 12, 14, 16, ...

2n + 8

Find the nth term (Write the explicit formula): -12, -2, 8, 18, ...

10n - 22

Find the 21st term in the sequence.

-4, 1, 6, 11, 16, ...

96

Find the 72nd term in the sequence:
-5, -1, 3, 7, ..

279

Find the first term in a sequence where the common difference is 6 and the eleventh term is 77.

17

Find the 12th term in a sequence with common difference -8 and first term 58.

-30

Find the first, second, fifth, and ninth terms of the sequence.
A(𝑛) = -9 + (𝑛-1)6

Write the first five terms of the sequence, explain what the fifth term means in context to the situation.
A baby's birth weight is 7 lbs. 4 oz., the baby gains 5 oz. each week.

Write the first five terms of the sequence, explain what the fifth term means in context to the situation.
The balance of a car loan starts at $4,500 and decreases $150 each month.

Find the second, fourth, and eighth terms of the sequence.
A(𝑛) = 1/5 + (𝑛 - 1)4/5

Is the pattern an arithmetic sequence? Why?
2, 5, 8, 11, _______, _______

Yes, each number is the previous number plus 3

Is the pattern an arithmetic sequence? Why?
2, 4, 8, 16, _______, _______

No, each number is the previous number multiplied by 2

Is the pattern an arithmetic sequence? Why?
1, 4, 9, 16, _______, _______

No, each term is the square of the next consecutive integer, starting with 1.

Is the pattern an arithmetic sequence? Why?
3, 9, 27, 81, _______, _______

No, each term is the previous number multiplied by 3

Is the pattern an arithmetic sequence? Why?
9, 15, 21, 27, _______, _______

Yes, each term is the previous number plus 6

Is the pattern an arithmetic sequence? Why?
2, -4, 8, -16, _______, _______

No, each term is the previous number multiplied by -2.