Arithmetic Sequences

Arithmetic vs. Geometric Sequences

Arithmetic Sequence: A sequence with a common difference between consecutive terms. The pattern is based on addition or subtraction.
Geometric Sequence: A sequence with a common ratio between consecutive terms. The pattern is based on multiplication or division.

Examples

Arithmetic Sequence: 3, 7, 11, 15, 19, 23, 27
Geometric Sequence: 3, 6, 12, 24, 48, 96, 192

Common Difference (d) in Arithmetic Sequences

To find the common difference, subtract any term from its subsequent term.
Example: In the sequence 3, 7, 11, 15, …, the common difference $d = 7 - 3 = 4$ .

Common Ratio (r) in Geometric Sequences

To find the common ratio, divide any term by its preceding term.
Example: In the sequence 3, 6, 12, 24, …, the common ratio $r = \frac{6}{3} = 2$ .

Arithmetic Mean

The arithmetic mean is the average of two numbers: $\frac{a + b}{2}$ .
In an arithmetic sequence, the arithmetic mean of two terms gives the middle number in the sequence.

Examples

The arithmetic mean of 3 and 11 is $\frac{3 + 11}{2} = \frac{14}{2} = 7$ .
The arithmetic mean of 7 and 23 is $\frac{7 + 23}{2} = \frac{30}{2} = 15$ .

Geometric Mean

The geometric mean of two numbers is the square root of their product: $\sqrt{a \times b}$ .
In a geometric sequence, the geometric mean of two terms gives the middle number in the sequence.

Examples

The geometric mean of 3 and 12 is $\sqrt{3 \times 12} = \sqrt{36} = 6$ .
To find the geometric mean between 6 and 96:
$\sqrt{6 \times 96} = \sqrt{6 \times 6 \times 16} = \sqrt{36 \times 16} = 6 \times 4 = 24$

Formula for the nth Term of a Sequence

Arithmetic Sequence

The formula to find the $n^{th}$ term (an$) of an arithmetic sequence is: $an = a_1 + (n - 1)d$ where:
- $a_1$ is the first term.
- $n$ is the term number.
- $d$ is the common difference.

Example

Find the 5th term of the arithmetic sequence 3, 7, 11, 15, …
- $a_1 = 3$
- $n = 5$
- $d = 4$
So, $a_5 = 3 + (5 - 1) \times 4 = 3 + 4 \times 4 = 3 + 16 = 19$ .

Geometric Sequence

The formula for finding the $n^{th}$ term (an) in a geometric sequence is: an = a1^(n−1)

Example

Calculate the 6th term of the geometric sequence 3, 6, 12, 24, 48, …
- $a_1 = 3$
- $r = 2$
- $n = 6$
So, $a_6 = 3 \times 2^{(6 - 1)} = 3 \times 2^5 = 3 \times 32 = 96$ .

Partial Sum of a Sequence

Arithmetic Sequence

The partial sum ( $Sn$ ) of the first n terms of an arithmetic sequence is given by: $Sn = \frac{(a1 + an)}{2} \times n$
This formula calculates the sum by taking the average of the first and last terms and multiplying by the number of terms.

Example

Find the sum of the first 7 terms of the arithmetic sequence 3, 7, 11, 15, 19, 23, 27.
- $a_1 = 3$
- $a_7 = 27$
- $n = 7$
$S_7 = \frac{(3 + 27)}{2} \times 7 = \frac{30}{2} \times 7 = 15 \times 7 = 105$

Geometric Sequence

The partial sum ( $Sn$ ) of the first n terms of a geometric sequence is given by: $Sn = a_1 \times \frac{1 - r^n}{1 - r}$
Where:
- $a_1$ is the first term.
- $r$ is the common ratio.
- $n$ is the number of terms.

Example

Find the sum of the first 6 terms of the geometric sequence 3, 6, 12, 24, 48, 96.
- $a_1 = 3$
- $r = 2$
- $n = 6$
$S_6 = 3 \times \frac{1 - 2^6}{1 - 2} = 3 \times \frac{1 - 64}{-1} = 3 \times \frac{-63}{-1} = 3 \times 63 = 189$

Sequences vs. Series

Sequence: A list of numbers (e.g., 3, 7, 11, 15, 19).
Series: The sum of the numbers in a sequence (e.g., 3 + 7 + 11 + 15 + 19).

Types of Sequences and Series

Finite Sequence/Series: Has a beginning and an end (e.g., 3, 7, 11, 15, 19).
Infinite Sequence/Series: Continues indefinitely (e.g., 3, 7, 11, 15, 19, …).
- The presence of dots indicates that the sequence or series goes on to infinity.

Practice Problems: Identifying Sequences and Series

Classify each of the following as a sequence or series, finite or infinite, and arithmetic, geometric, or neither.

a. 4, 7, 10, 13, 16, 19:
- Sequence
- Finite
- Arithmetic (common difference of 3)
b. 4, 8, 16, 32 …:
- Sequence
- Infinite
- Geometric (common ratio of 2)
c. 5 + 9 + 13 + 17 …:
- Series
- Infinite
- Arithmetic (common difference of 4)
d. 2 + 6 + 18 + 54 + 162:
- Series
- Finite
- Geometric (common ratio of 3)
e. 50, 46, 42, 38 …:
- Sequence
- Infinite
- Arithmetic (common difference of -4)
f. 3 + 12 + 48 …:
- Series
- Infinite
- Geometric (common ratio of 4)
g. 12 + 18 + 24 + 30 + 36:
- Series
- Finite
- Arithmetic (common difference of 6)

Writing Terms of a Sequence

Write the first four terms of the sequence defined by the formula $a_n = 3n - 7$ .
- $a_1 = 3(1) - 7 = -4$
- $a_2 = 3(2) - 7 = -1$
- $a_3 = 3(3) - 7 = 2$
- $a_4 = 3(4) - 7 = 5$
- The first four terms are -4, -1, 2, 5.

Finding the Next Terms in an Arithmetic Sequence

Write the next three terms of the arithmetic sequence 15, 22, 29, 36, …
- Find the common difference: $d = 22 - 15 = 7$ .
- Add the common difference to the last term to find the next terms:
  - $36 + 7 = 43$
  - $43 + 7 = 50$
  - $50 + 7 = 57$

Writing the First Few Terms of an Arithmetic Sequence

Write the first five terms of an arithmetic sequence given $a_1 = 29$ and $d = -4$ .
- The first term is 29.
- Add the common difference to generate subsequent terms:
  - $29 + (-4) = 25$
  - $25 + (-4) = 21$
  - $21 + (-4) = 17$
  - $17 + (-4) = 13$

Recursive Formulas

Write the first five terms of the sequence defined by the following recursive formulas.

Part a

$a1 = 3$ , $an = a_{n-1} + 4$
- $a2 = a1 + 4 = 3 + 4 = 7$
- $a3 = a2 + 4 = 7 + 4 = 11$
- $a4 = a3 + 4 = 11 + 4 = 15$
- $a5 = a4 + 4 = 15 + 4 = 19$

Part b

$a1 = 2$ , $an = 3 \times a_{n-1} + 2$
- $a2 = 3 \times a1 + 2 = 3 \times 2 + 2 = 8$
- $a3 = 3 \times a2 + 2 = 3 \times 8 + 2 = 26$
- $a4 = 3 \times a3 + 2 = 3 \times 26 + 2 = 80$
- $a5 = 3 \times a4 + 2 = 3 \times 80 + 2 = 242$

Writing General Formulas

Write a general formula (or explicit formula) for the sequences shown below.

Part a

8, 14, 20, 26, …
- The sequence is arithmetic.
- $a_1 = 8$
- $d = 14 - 8 = 6$
- $an = a1 + (n - 1)d = 8 + (n - 1)6 = 8 + 6n - 6 = 6n + 2$

Part b

$\frac{2}{3}, \frac{3}{5}, \frac{4}{7}, \frac{5}{9}, \frac{6}{11}, …$
- Separate the numerator and denominator into two sequences.
- Numerator: 2, 3, 4, 5, 6, … Arithmetic with $a1 = 2, d = 1, an = n + 1$
- Denominator: 3, 5, 7, 9, 11, … Arithmetic with $a1 = 3, d = 2, an = 2n + 1$

Further Problems

Part a

5, 14, 23, 32 ….
- $a_1 = 5$
- $d = 9$
- $a_n = 5 + (n-1)9$
- $a_n = 5 + 9n - 9$
- $a_n = 9n - 4$

Part b

150, 143, 136, 129
- $a_1 = 150$
- $d = -7$
- $a_n = 150 + (n-1)(-7)$
- $a_n = 150 -7n + 7$
- $a_n = 157 - 7n$

Part c

Calculate the value of the tenth term of the sequence

$a_{10} = 9(10) - 4$
$a_{10} = 90 - 4$
$a_{10} = 86$

Part d

Calculate the value of the tenth term of the sequence

$a_{10} = 157 - 7(10)$
$a_{10} = 157 - 70$
$a_{10} = 87$

Part e

Find the sum of the first ten terms

$Sn = (a1 + a_n) / 2 * n$
$S_{10} = (5 + 86) / 2 * 10$
$S_{10} = (91) / 2 * 10$
$S_{10} = 455$

Part f

Find the sum of the first ten terms

$Sn = (a1 + a_n) / 2 * n$
$S_{10} = (150 + 87) / 2 * 10$
$S_{10} = (237) / 2 * 10$
$S_{10} = 1185$

Further Practise

Part a

Find the sum of the first three hundred natural numbers

$Sn = (a1 + a_n) / 2 * n$
$S_{300} = (1 + 300) / 2 * 300$
$S_{300} = (301) / 2 * 300$
$S_{300} = 45150$

Part b

Calculate the sum of all even numbers from two to 100 inclusive

$Sn = (a1 + an) / 2 * n$ $an = a_1 + (n-1)d$
Calculate n
$100 = 2 + (n-1)2$
$49 = n - 1$
$n = 50$
$S_{50} = (2 + 100) / 2 * 50$
$S_{50} = (102) / 2 * 50$
$S_{50} = 2550$

Part c

Try this one determine the sum of all odd integers from 20 to seventy six

$Sn = (a1 + an) / 2 * n$ $an = a_1 + (n-1)d$
Calculate n
$75 = 21 + (n-1)2$
$27 = n - 1$
$n = 28$
$S_{28} = (21 + 75) / 2 * 28$
$S_{28} = (96) / 2 * 28$
$$S_{28} = 1344"