Jam saves increasing amounts of money each month from her allowance. She saves PHP50 the first month, PHP100 the second month, PHP150 the third month, and PHP200 the fourth month. This pattern of increasing savings by PHP50 each month forms a sequence. The question asks how much Jam will save in the tenth month if she continues this pattern of savings.

2. A sequence is a function whose domain is the set of
natural numbers. It is a logical order or arrangement of
values defined by a general term or nth term.
Each element of a sequence is called a term of a
sequence.

3. Lets Learn!
Jam saves Php50 from her monthly allowance on the first
month, Php100 on the second month, Php150 on the third
month, Php200 on the fourth month. If she will save
continuously in this manner, how much will she save on the
tenth month?

4. Let us list the amount of her savings.
The amount of monthly savings of Jam forms a sequence which is written as
(50, 100, 150, 200, 250, 300, 350, ... 500).
Month 1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th
Amount
of savings
Php50 Php100 Php150 Php200 Php250 Php300 Php350
______ ______ ______

5. Example 1. The following are examples
of sequences.
a. 1, 3, 5 7, 9, ...
b. 2, 4, 6, 8, 10, ...
c. 3, -2, -7, -12, -17
d. 5, 8, 11, 14, 17

6. Each member or element is called as term. So, 1, 3, 5, 7, 9 are
terms of a sequence and denoted as a symbol of A1, A2, A3, ...
and so on.
In example number 1,
A1= 1, A2= 3, A3= 5, A4= 7, A5= 9.

7. Example 2. List the first three elements of the sequence
defined by f(x) = 3x+5.
Since the domain of a sequence function is the set of natural numbers, we
consider the first three elements of the set of natural numbers.
Thus, we have
If x = 1, f(1) = 3(1) + 5 = 8
If x = 2, f(2) = 3(2) + 5 = 11
If x = 3, f(3) = 3(3) + 5 = 14
The first three elements of f(x) = 3x+5 are 8, 11 and 14 and these are
the first three elements of the sequence function.

8. Example 3. Give the first five elements of the sequence defined by
f(x) = 3x – 10.
f(x) = 3x – 10 f(x) = 3x – 10 f(x) = 3x - 10
f(1) = 3(1) – 10 f(2) = 3(2) – 10 f(3) = 3(3) - 10
= 3 – 10 = -7 = 6 – 10 = -4 = 9 – 10 = -1
f(x) = 3x - 10 f(x) = 3x - 10
f(4) = 3(4) - 10 f(5) = 3(5) - 10
= 12 – 10 = 2 = 15 – 10 = 5
Therefore the first five elements of f(x) = 3x – 10 are, -7, -4, -1, 2, 5.

9. Example 4. Give the first five elements of the sequence
defined by 3n2 - 1
f(n) = 3n2 – 1
f(1) = 3(1)2 – 1
= 3(1) – 1
= 3 – 1
= 2
f(n) = 3n2 - 1
f(2) = 3(2)2 – 1
= 3(4) – 1
= 12 – 1
= 11
f(n) = 3n2 – 1
f(3) = 3(3)2 – 1
= 3(9) – 1
= 27 – 1
= 26
f(n) = 3n2 – 1
f(4) = 3(4)2 – 1
= 3(16) – 1
= 48 – 1
= 47
f(n) = 3n2 – 1
f(5) = 3(5)2 – 1
= 3(25) – 1
= 75 – 1
= 74

10. A sequence can be finite or infinite.
• A finite sequence is a sequence whose number of terms is discrete and
has a last term. Thus, 3 -2, -7, -12 -17 and -2, -5, -8, -11, -13 are finite
sequence. Since it has a last term of 17 and -13.
• An infinite sequence is sequence whose terms are continuous and
does not have a last term. An ellipsis or three dots (...) at the end of any
sequence signifies the continuity of the terms. So 1, 3, 5, 7, 9, ... and 2, 4,
6, 8, 10, ... are infinite sequence.