[Func-Num] Sequence and Series

📋 This is my note-taking from what I learned in the class “Math175-002 Functions & Number Systems”

Linear Growth and Arithmetic Sequence

Suppose that you earn $30 each day. If you start out with $100 in your savings, and you add the $30 to your savings every evening, how much will you have saved after 11 days?

Days	Amount($)	Analysis
0	100	100 + 1 x 30
1	130	100 + 2 x 30
2	160	100 + 3 x 30

There is a pattern in the growth of the amount. At the end of the 11th day, you will have:

100 + 11 x 30 = $430 saved.

This additive growth is called linear growth.

1. Arithmetic Sequence

In an arithmetic sequence, the number obtained by subtracting any two consecutive terms is a constant. This constant “d” is called the “common difference” where “d = t_n+1 - t_n”; subscript indicates order of term in the sequence.

Ex) 4, 9, 14, 19 … → d = 9 - 4 = 5

Example: 1. Arithmetic Sequence

1. State the common difference of arithmetic sequence: a) 25, 21, 17, …

d = t_n+1 - t_n

Answer
d = 21 - 25 = -4

1 - a. The general term of the arithmetic sequence

a, a + d, a + 2d, a + 3d, … is “t_n = a + (n - 1)d,” n is order of term in the sequence.

Ex) Determine the 30th term of the arithmetic sequence 4, 7, 10, …

d = t_n+1 - t_n

t_n = a + (n - 1)d

Solution
d = 7 - 4 = 3
t30 = 4 + (30 - 1)(3) = 4 + 29(3) = 91

Example: 1 - a. The general term of the arithmetic sequence

Determine the indicated term of the arithmetic sequence: 23, 34, 45, … 10th term.

Solution
d = 34 - 23 = 11
t10 = 23 + (10 - 1)(11) = 23 + 9(11) = 122

1 - b. The Sum of an Arithmetic Series

For the general arithmetic series “a + (a + d) + (a + 2d) + … + t_n,” the sum of the first n terms is:

• S_n = \({n} \over {2}\) (a + t_n) → Use if last term is given!
• S_n = \({n} \over {2}\) [2a + (n - 1)d]

Ex) 1. Determine the sum of the first 30th terms of the arithmetic series: 5 + 5.3 + 5.6 + …

S_n = \({n} \over {2}\) [2a + (n - 1)d]

a = 5, d = 0.3, n = 30

Solution
S30 = (30/2) [2(5) + (30 – 1)0.3] = 280.5

Ex) 2. Determine the sum of the arithmetic series “–4, –10, –16, …, -94”

t_n = a + (n - 1)d

nth term = -94, d = -6. So, what is n = ?

Solution
tn = a + (n - 1)d
-94 = -4 + (n - 1)(-6)
- 94 + 4 = -6n + 6
- 90 - 6 = -6n
-96 = -6n
-96/-6 = n
n = 16

S_n = \({n} \over {2}\) (a + t_n) → Use if last term is given!

We know that there are 16th terms in the series and that the 16th term is – 94.

Solution
S16 = 16/2(-4 - 94) = -784

Example: 1 - b. The Sum of an Arithmetic Series

1. Determine the sum of the first 8 terms of the arithmetic sequence: 1.24 + 1.28 + 1.32 + …

S_n = \({n} \over {2}\) [2a + (n - 1)d]

Solution
a = 1.24, d = 0.04, n = 8
S8 = (8/2)[2(1.24) + (8-1)(0.04)] = 11.04

1. For the arithmetic sequence, a last term in the sequence is given, what is its order (subscript n)? what is the sum of the of the sequence? Sequence: 2, 7, 12, 17, …, 187.

t_n = a + (n - 1)d

Solution
nth term = 187, d = 5, what is n?
187 = 2 + (n - 1)(5)
187 - 2 = 5n - 5
185 + 5 = 5n
190/5 = n
n = 38

S_n = \({n} \over {2}\) (a + t_n)

We know that there are 38 terms in the series and that the 38th term is 187.

Solution
S38 = 38/2(2 + 187) = 3591

Exponential Growth and Geometric Sequences

An electric heater is now worth $100. Its value increases by 9% every year. How much will the heater be worth after 8 years?

Year	Value ($)	Method	Analysis
0	100.00	100	100
1	109.00	100 + 0.09 x 100.00	100 x 1.091
2	118.81	109.00 + 0.09 x 109.00	100 x 1.092

Noticing the pattern in the growth of the value, the value of the heater after 8 years is: 100(1.09)⁸ = $199.26

This multiplicative growth is called exponential growth.

2. Geometric Sequence

In a geometric sequence, the quotient obtained by dividing any two consecutive terms is a constant. This constant “r is called the common ratio,” where r = \({t\_{n+1}} \over {t_n}\)

Ex) 1, 3, 9, 27, … → r = 3/1 = 3

Example: 2. Geometric Sequence

State r of geometric sequence: 3, -12, 48, -192, …

r = \({t\_{n+1}} \over {t_n}\)

Solution
r = -12/3 = -4

2 - a. General term of the geometric sequence

a, ar, ar², ar³, … is t_n = a r^n-1

Ex) Determine the 11th term of the geometric sequence “3, 6, 12, 24, …”

a = 3 and r = 6/3 = 2, n - 1 = 11 - 1 = 10

Solution
t11 = 3(2)10 = 3072

Example: 2 - a. General term of the geometric sequence

Write the general term for the geometric sequence: 1, -2, 4, -8, …

t_n = a r^n-1

a = 1, r = -2/1 = -2

t_n = 1 (-2)^n-1 = (-2)^n-1

2 - b. The Sum of a Geometric Series

For the general geometric series “a + ar + ar² + ar³ + …,” the sum of the first n terms is S_n = \({a(r^n - 1)} \over {r - 1}\) , where r ≠ 1.

Ex) Determine the sum of the first 8 terms of the geometric series: “6 + 24 + 96 + …”

a = 6, r = 24/6 = 4, n = 8

S_n = \({a(r^n - 1)} \over {r - 1}\)

Solution
S8 = 6(48 - 1)/(4 - 1) = 131,070

Example: 2 - b. The Sum of a Geometric Series

Determine the sum of the first 12 terms of the geometric series: “1 – 2 + 4 – 8 + …”

S_n = \({a(r^n - 1)} \over {r - 1}\)

a = 1, r = -2, n = 12

S₁₂ = \({1((-2)^(12) - 1)} \over {-2 - 1}\) = \({(4096 - 1)} \over {-3}\) = \({4095} \over {-3}\) = -1365

Exercise

eCentennial/Content/Course Documents/Math175 Supplement: Page 17 and 18 All!

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