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PRE-CALCULUS
UNIT II – MATHEMATICAL INDUCTION

LESSON 2.1. REVIEW OF SEQUENCE AND SERIES

Learning Outcomes
At the end of the lesson, the student is able to:
(1) illustrate a series; and
(2) differentiate a series from a sequence.

Recall the following definitions:

A sequence is a function whose domain is the set of positive integers or the set {1, 2, 3, . . . , n}.

A series represents the sum of the terms of a sequence.

If a sequence is finite, we will refer to the sum of the terms of the sequence as the series
associated with the sequence. If the sequence has infinitely many terms, the sum is defined
more precisely in calculus.

A sequence is a list of numbers (separated by commas), while a series is a sum of numbers (separated
1 1 1 1 1 1
by “+” or “−” sign). As an illustration, 1, − , , − is a sequence, and 1 − + − is its associated series.
2 3 4 2 3 4

The sequence with the nth term 𝑎𝑛 is usually denoted by {𝑎𝑛 }, and the associated series is given by

S = a1 + a 2 + a 3 + · · · + a n .
Example 2.1.1. Determine the first five terms of each defined sequence, and give their associated series.

(1) {2 − n} (3) {(−1)n}

(2) {1 + 2n + 3n2} (4) {1 + 2 +3+· · · + n}

An arithmetic sequence is a sequence in which each term after the first is obtained by adding
a constant (called the common difference) to the preceding term.

If the nth term of an arithmetic sequence is 𝑎𝑛 and the common difference is d, then
𝑎𝑛 = 𝑎1 + (𝑛 − 1)𝑑

The associated arithmetic series with 𝑛 terms is given by

𝑛(𝑎1 + 𝑎𝑛 ) 𝑛[2𝑎1 + (𝑛 − 1)𝑑]
𝑆𝑛 = =
2 2

A geometric sequence is a sequence in which each term after the first is obtained by
multiplying the preceding term by a constant (called the common ratio).

If the nth term of a geometric sequence is an and the common ratio is r, then
𝑎𝑛 = 𝑎1 𝑟 𝑛−1
The associated geometric series with n terms is given by
𝑛𝑎1 𝑖𝑓 𝑟 = 1
𝑛)
𝑎
𝑆𝑛 = { 1 (1 − 𝑟
𝑖𝑓 𝑟 ≠ 1
(1 − 𝑟)
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When −1 < r < 1, the infinite geometric series
a1 + a1r + a1r2 + · · · + a1rn−1 + · · ·
has a sum, and is given by
𝑎1
𝑆=
(1 − 𝑟)

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If { 𝑎1 } is an arithmetic sequence, then the sequence with nth term 𝑏𝑛 = is a
𝑎1
harmonic sequence.

Fibonacci sequence is a sequence in which each term is obtained by adding the two
preceding terms. 𝑎𝑛 = 𝑎𝑛−2 + 𝑎𝑛−1

Exercise 2.1.1
1. Write SEQ if the given item is a sequence, and write SER if it is a series.
(a) 1, 2, 4, 8, . . .
(b) 2, 8, 10, 18, . . .
(c) −1 + 1 − 1 + 1 − 1
1 2 3 4
(d) , , , , …
2 3 4 5
(e) 1 + 2 + 22 + 23 + 24
(f) 1 + 0.1 + 0.001 + 0.0001

2. Write A if the sequence is arithmetic, G if it is geometric, F if Fibonacci, and O if it is not one of the
mentioned types.
1 1 1 1 1
(a) 3, 5, 7, 9, 11, . . . (e) , , , , ,…
5 9 13 17 21
(b) 2, 4, 9, 16, 25, . . . (f) 4, 6, 10, 16, 26, . . .
1 1 1 1
(c) , , , ,… (g) √3, √4, √5, √6, …
4 16 64 256
1 2 3 4
(d) , , , , … (h) 0.1, 0.01, 0.001, 0.0001, . . .
3 9 27 81

3. Determine the first five terms of each defined sequence, and give their associated series.
(a) {1 + n − n2} (c) a1 = 3 and an = 2an-1 + 3 for n ≥ 2
(b) {1 − (−1)n+1} (d) {1 · 2 · 3 · · · n}
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LESSON 2.2. SIGMA NOTATION
Learning Outcomes
At the end of the lesson, the student is able to:
(1) definition of and writing in sigma notation
(2) evaluate sums written in sigma notation
(3) properties of sigma notation
(4) calculating sums using the properties of sigma notation

The sigma notation is a shorthand for writing sums. In this lesson, we will see the power of this notation
in computing sums of numbers as well as algebraic expressions.

2.2.1. Writing and Evaluating Sums in Sigma Notation

Mathematicians use the sigma notation to denote a sum. The uppercase Greek letter Σ (sigma) is used
to indicate a “sum.” The notation consists of several components or parts.

Let 𝑓(𝑖) be an expression involving an integer 𝑖. The expression

𝑓(𝑚) + 𝑓(𝑚 + 1) + 𝑓(𝑚 + 2) + ⋯ + 𝑓(𝑛)
can be compactly written in sigma notation, and we write it as
𝑛

∑ 𝑓(𝑖),
𝑖=𝑚

which is read “the summation of 𝑓(𝑖) from 𝑖 = 𝑚 to 𝑛.” Here, 𝑚 and 𝑛 are integers with 𝑚 ≤
𝑛, 𝑓(𝑖) is a term (or summand) of the summation, and the letter 𝑖 is the index, 𝑚 the lower
bound, and 𝑛 the upper bound.

Example 2.2.1. Expand each summation, and simplify if possible.

4
(1) ∑(2𝑖 + 3)
𝑖=2
5
(2) ∑ 2𝑖
𝑖=0
𝑛
(3) ∑ 𝑎𝑖
𝑖=1
6
√𝑛
(4) ∑
𝑛+1
𝑛=1

Example 2.2.2. Write each expression in sigma notation.

1 1 1 1
(1) 1 + + + + ⋯ +
2 3 4 100
(2) − 1 + 2 − 3 + 4 − 5 + 6 − 7 + 8 − 9 + ⋯ − 25
(3) 𝑎2 + 𝑎4 + 𝑎6 + 𝑎8 + ⋯ + 𝑎20
1 1 1 1 1 1 1
(4) 1 + + + + + + +
2 4 8 16 32 64 128
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Seatwork/Homework 2.2.1
1. Expand each summation, and simplify if possible.
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(𝑎) ∑ (2 − 3𝑘)
𝑘=−1
𝑛
(𝑏) ∑ 𝑥 𝑗
𝑗=1
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(𝑐) ∑(𝑗 2 − 𝑗)
𝑗=3
4
(𝑑) ∑(−1)𝑘+1 𝑘
𝑘=1
3
(𝑒) ∑(𝑎𝑛+1 − 𝑎𝑛 )
𝑛=1
2. Write each expression in sigma notation.
(𝑎) 𝑥 + 2𝑥 2 + 3𝑥 3 + 4𝑥 4 + 5𝑥 5
(𝑏) 1 − 2 + 3 − 4 + 5 − 6 + ⋯ − 10
(𝑐) 1 + 3 + 5 + 7 + ⋯ + 101
(𝑑) 𝑎4 + 𝑎8 + 𝑎12 + 𝑎16
1 1 1 1
(𝑒) 1 − + − +
3 5 7 9

2.2.2. Properties of Sigma Notation

We start with finding a formula for the sum of

𝑛

∑𝑖 = 1 +2 +3 +⋯+𝑛
𝑖=1
in terms of 𝑛.

𝑛 𝑛

∑ 𝑐 𝑓(𝑖) = 𝑐 ∑ 𝑓(𝑖), 𝑐 any real number.

𝑖=𝑚 𝑖=𝑚

𝑛 𝑛 𝑛

∑[𝑓(𝑖) + 𝑔(𝑖)] = ∑ 𝑓(𝑖) + ∑ 𝑔(𝑖)

𝑖=𝑚 𝑖=𝑚 𝑖=𝑚
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𝑛

∑ 𝑐 = 𝑐(𝑛 − 𝑚 + 1)
𝑖=𝑚

A special case of the above result which you might encounter more often is the following:
𝑛

∑ 𝑐 = 𝑐𝑛
𝑖=1

Telescoping Sum
𝑛

∑[𝑓(𝑖 + 1) − 𝑓(𝑖)] = 𝑓(𝑛 + 1) − 𝑓(𝑚)

𝑖=𝑚

Example 2.2.3. Evaluate:

30

∑(4𝑖 − 5)
𝑖=1

Example 2.2.4. Evaluate:

1 1 1 1
+ + +⋯+
1∙2 2∙3 3∙4 99 ∙ 100

Seatwork/Homework 2.2.2
1. Use the properties of sigma notation to evaluate the following summations.
50
(𝑎) ∑(2 − 3𝑘)
𝑘=1
𝑛
(𝑏) ∑(1 + 2𝑗)
𝑗=1

99
1
(𝑐) ∑
𝑗=1
√𝑖 + 1 + √𝑖

2. If ∑𝑛𝑖=1(𝑖 + 1)2 = 𝑎𝑛3 + 𝑏𝑛2 + 𝑐𝑛 + 𝑑, what is 𝑎 + 𝑏 + 𝑐 + 𝑑?