The document defines sequences and series in precalculus. A sequence is a function with positive integers as its domain. A series represents the sum of the terms of a sequence. The document provides examples of arithmetic and geometric sequences, and defines their associated series. It also discusses infinite geometric series and harmonic sequences. Examples are given to identify sequences and series, determine sequence terms, and identify types of sequences.

1. Sequencesand
Series
PreCalculus

2. 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.

3. Sequences and series
The sequence with nth term an is usually
denoted by {an}, and the associated series is
given by S = a1 + a2 + a3 + ··· + an.

4. Determine the first five terms of each defined
sequence, and give their associated series.
(1) {2 - n}
(2) {1+2n + 3n2}
(3) {(-1)n}
(4) {1+2+3+ ··· + n}

5. Determine the first five terms of each defined
sequence, and give their associated series.
(2) {1+2n + 3n2}
(3) {(-1)n}
(4) {1+2+3+ ··· + n}

6. Determine the first five terms of each defined
sequence, and give their associated series.
(3) {(-1)n}
(4) {1+2+3+ ··· + n}

7. Determine the first five terms of each defined
sequence, and give their associated series.
(4) {1+2+3+ ··· + n}

8. Arithmetic sequences and series
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

9. Arithmetic sequences and series
The associated arithmetic series with n terms is given by

10. Geometric sequences and series
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

11. Geometric sequences and series
The associated geometric series with n terms is given by
r
ra
S
n
n



1
)1(1
Where r is not equal to zero.

12. Infinite geometric series
When -1 < r < 1, the infinite geometric series
a1 + a1r + a1r2 + ··· + a1rn-1 + ···
has a sum, and is given by

13. Harmonic sequence
If {an} is an arithmetic sequence, then the
sequence with nth term bn =
1
𝑎 𝑛
is a harmonic
sequence.

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

15. Seatwork
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.
(a) 3, 5, 7, 9, 11,... Answer: A (e) 1 5 , 1 9 , 1 13 , 1 17 , 1 21 ,...
(b) 2, 4, 9, 16, 25,... Answer: O (f) 4, 6, 10, 16, 26,...
(c) 1 4 , 1 16 , 1 64 , 1 256 ,... Answer: G (g) p3, p4, p5, p6,...
(d) 1 3 , 2 9 , 3 27 , 4 81 ,... Answer: O (h) 0.1, 0.01, 0.001, 0.0001,...
a A, b O, c G, d O, e A ,f F, g O ,h G

16. Seatwork
3. Determine the first five terms of each defined sequence,
and give their associated series.
(a) {1 + n - n2}
(b) {1 -(-1)n+1}
(c) a1 = 3 and an = 2an-1 + 3 for n≥2
(d) {1 · 2 · 3 ··· n}

17. Seatwork
4. Identify the series (and write NAGIG if it is not arithmetic, geometric, and infinite geometric
series), and determine the sum (and write NO SUM if it cannot be summed up).

18. Reference:
Garces, Ian June L. et. al. (2016) Precalculus “Teaching Guide for
Senior High School, Commission on Higher Education

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