Learn – Algebra foundations – Sequences

Lesson notes
Problem set 1
Problem set 2
Problem set 3

Lesson notes

An arithmetic sequence or an arithmetic progression is a sequence of numbers such that the difference between any two consecutive numbers in the sequence is the same. Examples of arithmetic sequences include:
1. $2,3,5,8,\cdots$
2. $10, 1, -8, -17, -26, \cdots$
3. $-3,-4.5,-6,-7.5, \cdots$
A geometric sequence or a geometric progression is a sequence of numbers such that the ratio between any two consecutive numbers in the sequence is the same. Examples of geometric sequences include:
1. $3,6,12,24,48\cdots$
2. $10, 0.1, 0.01, 0.001, \cdots$

Problem set 1

Determine the hundredth term in the sequence: $2, 1.9, 1.8, 1.7, 1.6, 1.5,\cdots$
Determine the tenth term in the sequence: $3, \frac{3}{2}, \frac{3}{4}, \frac{3}{8}, \frac{3}{16}, \cdots$
Determine the fifteenth term in the sequence: $2, \frac{3}{2}, \frac{4}{4}, \frac{5}{8}, \frac{6}{16},\cdots$
For an arithmetic sequence, assume $a$ is the first term, and $d$ is the common difference between consecutive terms. Derive a formula to determine the $n$ th term of the progression in terms of $a, d$ and $n$ .
For a geometric sequence, assume $a$ is the first term, and $r$ is the common ratio between consecutive terms. Derive a formula to determine the $n$ th term of the progression in terms of $a, r$ and $n$ .

Problem set 2

The $14\textsuperscript{th}$ and $34\textsuperscript{th}$ terms of an arithmetic sequence are $18$ and $48$ respectively. What is the sum of the sequence from its $10\textsuperscript{th}$ term through its $40\textsuperscript{th}$ term?
The sum of an arithmetic sequence from its $300\textsuperscript{th}$ term through its $399\textsuperscript{th}$ term is $67.89$ . The common difference $d$ for the sequence is $0.00345$ . What is the sum of the arithmetic sequence from its $100\textsuperscript{th}$ term through its $199\textsuperscript{th}$ term?
In a geometric sequence, the $200\textsuperscript{th}$ and the $300\textsuperscript{th}$ terms are $20$ and $30$ respectively. What is the $400\textsuperscript{th}$ term of the sequence?
$1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8},\cdots$ is a geometric sequence. If you add the terms of the sequence starting from the very first term, would the sum ever become bigger than $2$ ? How many terms must you add to get the sum to be bigger than $1.9999$ ? (You are allowed to use calculator for this problem)

Problem set 3

A geometric sequence has $a$ as its first term and $r$ as its common ratio. Determine a formula for the product of first $n$ terms of the sequence in terms of $a$ , $r$ and $n$ .
For an arithmetic progression, assume $a$ is the first term, and $d$ is the common difference between consecutive terms. Derive a formula to determine the sum of the first $n$ terms of the progression in terms of $a, d$ and $n$ .
For a geometric progression, assume $a$ is the first term, and $r$ is the common ratio between consecutive terms. Derive a formula to determine the sum of the first $n$ terms of the progression in terms of $a, r$ and $n$ .
$a, a+d, a+2d, a+3d, \cdots$ and $b, b+e, b+2e, b+3e, \cdots$ are two arithmetic sequences. What can you say about the new sequence formed by adding the terms of the sequences? What is the sum of the first $n$ terms of the new sequence?
$a, ar, ar^2, ar^3, \cdots$ and $b, bs, bs^2, bs^3, \cdots$ are two geometric sequences. What can you say about the new sequence formed by dividing the terms of the first sequence by the terms of the second sequence? What is the sum of the first $n$ terms of the new sequence?