Sequences and Series

Grade 12 Math - Sequences and Series

Lesson Objectives

Define and differentiate arithmetic and geometric progressions.
Calculate any term in an arithmetic or geometric sequence.
Determine the sum of a specified number of terms in AP or GP.
Solve intermediate to difficult real-world problems involving AP and GP.
Identify patterns in sequences and use formulas to solve problems.

Lesson Introduction

Sequences and series are foundational concepts in mathematics that describe patterns. You see these in saving plans, interest calculations, and even in nature. In this lesson, we'll explore arithmetic and geometric progressions—how they are formed, how to find specific terms, and how to sum them efficiently.

Core Lesson Content

Arithmetic Progression (AP) is a sequence in which the difference between any two consecutive terms is constant.

Formula for the nth term of an AP:

$a_n = a + (n - 1)d$

Sum of the first n terms of an AP:

$S_n = \frac{n}{2}(2a + (n - 1)d)$

Geometric Progression (GP) is a sequence in which each term after the first is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio.

Formula for the nth term of a GP:

$a_n = ar^{n - 1}$

Sum of the first n terms of a GP:

$S_n = \frac{a(1 - r^n)}{1 - r}, \quad \text{for } r \ne 1$

Sum to infinity of a GP (if $|r| < 1$ ):

$S = \frac{a}{1 - r}$

Worked Example

Example 1: Find the 10th term of the AP: 3, 7, 11, 15,...

$a = 3,\ d = 4,\ n = 10$

$a_{10} = 3 + (10 - 1) \cdot 4 = 3 + 36 = 39$

Example 2: Find the sum of the first 20 terms of the AP: 5, 10, 15,...

$a = 5,\ d = 5,\ n = 20$

$S_{20} = \frac{20}{2}(2 \cdot 5 + (20 - 1) \cdot 5) = 10(10 + 95) = 10 \cdot 105 = 1050$

Example 3: If the 2nd term of a geometric progression is 12 and the 5th term is 96, find the first term and the common ratio.
The general form for the nth term of a geometric progression is:

a_n = ar^{n - 1}

For the 2nd term and the 5th term, we can write:

ar^{2 - 1} = 12

(equation 1)

ar^{5 - 1} = 96

(equation 2)

Simplifying the equations:
Equation 1:

ar = 12

Equation 2:

ar^4 = 96

Divide equation 2 by equation 1 to eliminate

a

\frac{ar^4}{ar} = \frac{96}{12} \quad \Rightarrow \quad r^3 = 8 \quad \Rightarrow \quad r = 2

Now substitute

r = 2

into equation 1 to find

a

a \cdot 2 = 12 \quad \Rightarrow \quad a = 6

Answer: The first term

a = 6

and the common ratio

r = 2

Example 4: Find the sum of the first 6 terms of the GP: 3, 6, 12,...

$a = 3,\ r = 2,\ n = 6$

$S_6 = \frac{3(1 - 2^6)}{1 - 2} = \frac{3(1 - 64)}{-1} = 189$

Example 5: What is the sum to infinity of the GP: 16, 8, 4, 2,...

$a = 16,\ r = \frac{1}{2}$

$S = \frac{16}{1 - \frac{1}{2}} = 32$

Example 6: How many terms of the AP: 7, 10, 13,... will sum up to 370?

$S_n = \frac{n}{2}(2 \cdot 7 + (n - 1) \cdot 3) = 370$

Solve: $n(3n + 11) = 740 \Rightarrow 3n^2 + 11n - 740 = 0$

Example 7: Find the 12th term of the AP: -5, -2, 1,...

$a = -5,\ d = 3,\ n = 12$

$a_{12} = -5 + (12 - 1) \cdot 3 = 28$

Example 8: Determine the number of terms for which the sum of the AP: 4, 8, 12,... equals 600.

$S_n = \frac{n}{2}(2 \cdot 4 + (n - 1) \cdot 4) = 600$

$\frac{n}{2}(4n + 4) = 600 \Rightarrow n(n + 1) = 300$

Example 9: Find the sum of the GP: 10, -5, 2.5,... up to 5 terms.

$a = 10,\ r = -\frac{1}{2},\ n = 5$

$S_5 = \frac{10(1 - (-\frac{1}{2})^5)}{1 + \frac{1}{2}} = 6.875$

Example 10: If the 3rd term of a geometric progression is 16 and the 6th term is 128, find the first term and the common ratio.

$ar^2 = 16,\ ar^5 = 128$

$r = 2,\ a = 4$

Exercises

Find the 15th term of the AP: 6, 9, 12,...
Find the sum of the first 30 terms of the AP: 2, 4, 6,...
If the 3rd term of a geometric progression is 16 and the 6th term is 128, find the first term and the common ratio.
[WAEC] Find the term of the AP: 20, 16, 12,... that equals -20. (Past Question)
[NECO] Find the sum of the first 8 terms of the GP: 2, 4, 8,... (Past Question)
Simplify and find the sum to infinity of GP: 5, 2.5, 1.25,...
[JAMB] How many terms of the AP: 3, 6, 9,... will sum to 333? (Past Question)
Find the 10th term of a GP where $a = 10,\ r = 0.1$
Determine the number of terms required for the AP: 4, 7, 10,... to reach a sum of 253.
[WAEC] Find the sum of the GP with $a = 6,\ r = -\frac{1}{2},\ n = 5$ (Past Question)

Conclusion/Recap

In this lesson, you've learned how to calculate specific terms and sums of arithmetic and geometric progressions. These skills are vital in real-world financial modeling, engineering, and science.