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)249 = 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)227 = n - 1
n = 28
S_{28} = (21 + 75) / 2 * 28
S_{28} = (96) / 2 * 28
$$S_{28} = 1344"