PPT_SequenceAndSeries.pdf

Sequence and
Series
STEM-Belardo

Sequence and series are two related
terms in Mathematics. Both involve
patterns of numbers. Sequence
shows the listing of these numbers
while series expresses the
associated sum of the sequence.
SEQUENCE and SERIES

A sequence is a function whose domain is
the set of positive integers or the set of
counting numbers, which is {1,2,3,…,𝑛}.
For example, 2, 4, 6, 8 is a sequence since
2 can be expressed as 2(1), 4 as 2(2), 6 as 2(3),
and 8 as a 2(4).
SEQUENCE

The general form of a sequence is 𝑎1,𝑎2, 𝑎3,…where
each numerical subscript denotes the term in the
sequence.
Each element of the sequence is called term.
The 𝑛𝑡ℎ term of asequence is denoted by𝑎𝑛.
It can be represented by amathematical rule, f(n) =𝑎𝑛
SEQUENCE

A series represents the sum of the terms of a sequence. It is
usually expressed with “+” or “ – ” sign in between the terms. If
a sequence is finite, the sum of the terms of the sequence is
referred to as the series associated with the sequence.
In the example, 2, 4, 6, 8, the series associated with the
sequence is 2 + 4 + 6 + 8 which is equal to 20.
The associated series of a sequence is defined by
S=𝑎1 +𝑎2 +𝑎3 +𝑎4 +…+𝑎𝑛.
SERIES

1.Josewants toincreasehervocabulary. OnMonday he
learned themeanings offive newwords.Each otherdaythat
week,heincreased the numberofnewwordsthat helearned
bythree.
1.Writethesequenceforthenumber ofnewwords that
Joselearned eachdayforaweek.
2.Express theassociated seriesofthesequence.
3.Writethemathematical rule that couldgenerate all
theterms ofthesequence.
SEQUENCE and SERIES
5,8,11,14,17,20,23
𝑆7 =5+8+11+14+17+20+23=98
𝑆𝑛 =3n+2

An arithmetic sequence is a
list of numbers with a common
difference between consecutive
terms.
ARITHMETIC SEQUENCE and SERIES

For example,
the sequence 4,6, 8,10, ...
The sequence 20, 15, 10, 5,...
The sequence 1,2, 4,8 ...
Arithmetic sequence
Arithmetic sequence
NOT arithmetic sequence

General Term
𝑎𝑛 = 𝑎1 + 𝑛 − 1 𝑑
𝑛𝑡ℎ 𝑡𝑒𝑟𝑚
𝑐𝑜𝑚𝑚𝑜𝑛 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒
𝑓𝑖𝑟𝑠𝑡 𝑡𝑒𝑟𝑚
𝑡𝑒𝑟𝑚 𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛

𝑎𝑛 = 𝑎1 + 𝑛 − 1 𝑑
𝑎20 = 100 + 20 − 1 (−3)
𝑎20 = 100 + 19 (−3)
𝑎20 = 100 − 57
𝑎20 = 43
1. 100,97,94,91,…20th term
d=-3 n=20 𝑎1 = 100 𝑎20=?

𝑎𝑛 = 𝑎1 + 𝑛 − 1 𝑑
𝑎15 = 2 + 15 − 1 (5)
𝑎15 = 2 + 14 (5)
𝑎15 = 2 + 70
𝑎15 = 72
2. 2, 7, 12, 17, … 15th term
d=5 n=15 𝑎1 = 2 𝑎15=?

An arithmetic series is the indicated sum of
the terms of an arithmetic sequence.
The associated arithmetic series (𝑆𝑛) with
n terms is given by:

Find the associated arithmetic series of each given sequence.
-10, -20, -30, -40, …, -110
𝑎1 = −10 𝑎𝑛 = −110 𝑑 = −10 𝑛 =?
𝑎𝑛 = 𝑎1 + 𝑛 − 1 𝑑
−110 = −10 + 𝑛 − 1 (−10)
−100 = −10𝑛 + 10
10𝑛 = 100 + 10
10𝑛 = 110
𝑛 = 11

𝑆11 =
11(−120)
2
-10, -20, -30, -40, …, -110
𝑆𝑛 =
𝑛(𝑎1 + 𝑎𝑛)
2
𝑆11 =
11(−10−110)
2
𝑆11 = −660

3, 9, 15, 21, … up to 20th term
𝑎1 = 3 𝑑 = 6 𝑛 = 20
𝑆20 =
𝑛 2𝑎1 + 𝑛 − 1 𝑑
2
𝑆20 =
20 2(3) + 20 − 1 (6)
2
𝑆20 =
20 6 + (19)(6)
2
𝑆20 =
20 6 + 114
2
𝑆20 =
20(120)
2
𝑆20 = 1 200

Cesar is creating a program. On the first day, he made 5 lines of codes. As he
becomes skilled in writing codes, he writes one more line than the previous
day. He finishes the program on the 6th day. How many lines of codes did he
write?
𝑎1 = 5 𝑑 = 1 𝑛 = 6
𝑆𝑛 =
𝑛 2𝑎1 + 𝑛 − 1 𝑑
2
𝑆6 =
6 2(5) + 6 − 1 (1)
2
𝑆6 =
6 10 + 5 (1)
2
𝑆6 =
6(15)
2
= 45

A geometric sequence is a
sequence in which each term
after the first is obtained by
multiplying the preceding term
by a constant.
GEOMETRIC SEQUENCE and SERIES

The sequence 2,4, 8,16, 32, … is ageometric sequence
because…
The sequence -3, 9, -27, 81, … is ageometric sequence
because…
The sequence 5,7, 9,11, …is not a geometric sequence
because…

The constant ratio between consecutive entries of
a geometric sequence is called a common ratio,
denoted by r.
Generally, the 𝑛𝑡ℎ term of geometric sequence is given
by
𝑎𝑛 = 𝑎1𝑟𝑛−1
𝑙𝑎𝑠𝑡 𝑡𝑒𝑟𝑚
𝑓𝑖𝑟𝑠𝑡 𝑡𝑒𝑟𝑚
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑒𝑟𝑚𝑠
𝑐𝑜𝑚𝑚𝑜𝑛 𝑟𝑎𝑡𝑖𝑜

Find the common ratio and the 𝑛𝑡ℎ
term.
5, 10, 20, 40, 80, …10𝑡ℎ
term
The common ratio is 2.
𝑎10 = (5)(2)10−1
𝑎10 = (5)(2)9
𝑎10 = (5)(2)9
𝑎10 = (5)(512) 𝑎10 = 2 560

Find the common ratio and the 𝑛𝑡ℎ
term.
-4, -12, -36, -108, …12𝑡ℎ
term
The common ratio is 3.
𝑎12 = (−4)(3)12−1
𝑎12 = (−4)(3)11
𝑎12 = (−4)(177 147)
𝑎12 = −708 588

7𝑡ℎ
𝑡𝑒𝑟𝑚: −6
6𝑡ℎ 𝑡𝑒𝑟𝑚: 2
5𝑡ℎ
𝑡𝑒𝑟𝑚: −
2
3

−6=𝑎1(−3)6
−6=𝑎1(729)
−6=𝑎1(729)
−6
729
= 𝑎1
−2
243
= 𝑎1

𝑎5 =
−2
243
(−3)4
𝑎5 =
−2
243
(81)
𝑎5 =
−2(81)
243
𝑎5 =
−2
3

A geometric series is the indicated sum of the terms of a geometric
sequence. The associated geometric series (𝑆𝑛) with n terms is given by:
𝐶𝑎𝑠𝑒 1, 𝑟 = 1
𝑆𝑛=n𝑎1
𝐶𝑎𝑠𝑒 2, 𝑟 ≠ 1
𝑆𝑛 =
𝑎1(1−𝑟𝑛)
(1−𝑟)
𝐶𝑎𝑠𝑒 3, −1 < 𝑟 < 1
𝑆 =
𝑎1
1−𝑟

Letn=45,𝑎1 =7
Find the associated geometric series of each given sequence.
7, 7, 7, 7, …., up to 45𝑡ℎ term
Since r = 1, case 1 will be applied.
𝐶𝑎𝑠𝑒 1,
𝑆𝑛 = 𝑛𝑎1
𝑆45 = (45)(7)
𝑆45 = 315

Let𝑛 = 10, 𝑎1 = 5, 𝑟 = 2
5, 10, 20, 40, 80, … 10𝑡ℎ term
𝐶𝑎𝑠𝑒 2, 𝑟 ≠ 1
𝑆𝑛 =
𝑎1(1−𝑟𝑛)
(1−𝑟)
𝑆10 =
(5)(1−1024)
−1
𝑆10 =
(5) 1−(2)10
(1−2)
𝑆10 =
(5)(−1023)
−1
𝑆10 = 5 115

Let 𝑎1 = 1, 𝑟 =
1
3
1,
1
3
,
1
9
,
1
27
, …
Since r =
1
3
, case 3 will be applied.
𝐶𝑎𝑠𝑒 3, −1 < 𝑟 < 1
𝑆 =
𝑎1
1−𝑟
𝑆 =
1
2
3
𝑆 =
1
1−
1
3
𝑆 =
3
2

Let𝑛 = 8, 𝑎1 = 10, 𝑟 = 2
If you get paid PhP 10.00 for the first hour, PhP 20.00 for the second
hour, PhP 40.00 for the third hour, how much is your total money at the
end of eight hours?
𝐶𝑎𝑠𝑒 2, 𝑟 ≠ 1
𝑆𝑛 =
𝑎1(1−𝑟𝑛)
(1−𝑟)
𝑆8 =
(10)(1−256)
−1
𝑆8 =
(10) 1−(2)8
(1−2)
𝑆8 =
(10)(−255)
−1
𝑆8 = 2550

A ball dropped from the top of a building 180 m high always rebounds
one-half the distance it has fallen. How far the ball has travelled before
coming to rest?

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