Sequences: Math 10 Q1 Reviewer
Patterns and Sequences
Pattern
A series or sequence that repeats
Sequence
A succession of numbers. Each number in a sequence has a term which are formed according
to a fixed rule or property
Formulas
Arithmetic Sequence (Explicit) Arithmetic Sequence (Recursive)
an = a1 + (n − 1)d an = an−1 + d
Arithmetic Means and Series
Arithmetic Means
Arithmetic Means are the terms in the middle of a sequence.
Formulas
The Common Difference from the Sequence’s Extremes
an − a1
d=
n−1
Arithmetic Series
The value of the sum of all the terms in the series.
Formulas
Arithmetic Sequence, n is known Arithmetic Sequence, n is unknown
n n
Sequences: Math 10 Q1 Reviewer 1
n n
Sn = (a1 + an ) Sn = [2a1 + (n − 1)d]
2 2
Geometric Sequence, Means, and Series
Geometric Sequence
A sequence which is obtained by multiplying the preceding term with a common ratio.
Formula
Geometric Sequence Common Ratio (Trad.) Common Ratio (Ed.)
an = a1 r n−1 ah ah
h−x rx =
ax ax
root index = h − x
*wherein h is the higher
positioned term and x is
the lower positioned term.
Geometric Means
Formulas
Common Ratio from Extremes
Use Geometric Sequence formula and isolate r .
Geometric Series
Formulas
Geometric Series, Finite Geometric Series, Infinite
a1 (1 − r n ) a1
Sn = S∞ =
1−r 1−r
Sequences: Math 10 Q1 Reviewer 2
Polynomials
A polynomial is an expression consisting of, variables and coefficients which are combined
through means of addition, subtraction and multiplication only.
What are and are not variables?
A algebraic expression is not a polynomial when:
1. It has a fractional exponent. 3. Has a fraction with a variable as the denominator.
1 6
Ex.: 3x 2 Ex.: 5x2 + 12x − x
2. Has a negative exponent. 4. Has a variable in a radical expression. Ex.:
Ex.: 10x−3 + 2 49x2 − 3x + 5
Theorems
Remainder Theorem
If the polynomial P (x) is divided by x − r , the remainder R is a constant and is equal to
P (r).
Or, P (r) =R
Factor Theorem
The Factor Theorem states that the dividend polynomial is a has the divisor as its factor if
and only if their remainder is zero.
Or, P (x) has x − r as a factor if and only if P (r) = 0.
Rational Root Theorem
Let an xn + an−1 xn−1 + ... + a1 x + a0 = 0 where an = 0, the possible rational
p
zeros of P (x) are in the form of q where p is a factor of ao and q is a factor of an .
Or:
p a factor of the last term (a0 )
=
q a factor of the first term (an )
Sequences: Math 10 Q1 Reviewer 3
Factorization
Reverse Distribution/Extraction
ex.: −6x5 − 15x4 + 9x3
This polynomial has terms with factors of −3x3 ,
−6x5 − 15x4 + 9x3
therefore, it can be extractable.
−3x3 (2x2 + 5x − 3)
However, this process is still not complete as
(2x2 + 5x − 3) is still factorable.
Factoring by Product of Two Binomials
A trinomial with a factor of 2 can always be shown as a product of two binomials. In
factoring, we must understand the following.
Looking at the factored equation, the terms must
ax2 + bx + c = 0
satisfy that the statement below.
(1−x + 1−)(1−x + 1−)
outer term′ s product + inner term′ s product = the middle term
The coefficients in with the x should be the
factors of a
Factoring by Synthetic Division
ex.: Continuation: −6x5 − 15x4 + 9x3
We can find for the dominant coefficients and see that −3∣ −2 5 −3
they’re divisible by 3. We can first extract 3x3 before −6 −3
dividing. 2 −1 0
Then, divide (2x2 + 5x − 3) to 3. 3 would then
(2x − 1)(x + 3)
become −3 because of (x − r).
−3x3 (2x − 1)(x + 3)
Special Products/Factors
Sequences: Math 10 Q1 Reviewer 4
Involving Squares:
Distributive Law ax + ay = a(x + y)
Difference of Two Squares (x + y)(x − y) = x2 − y2
Square of a Sum x2 + 2xy + y2 = (x + y)2 = (x + y)(x + y)
Square of a Difference x2 −2xy + y2 = (x−y)2 = (x − y)(x − y)
Involving Cubes:
Cube of a Sum x3 + 3x2 y + 3xy2 + y3 = (x + y)3 = (x + y)(x + y)(x +
y)
Cube of a Difference x3 − 3x2 y + 3xy2 − y3 = (x − y)3 = (x − y)(x −
y)(x − y)
Sum of Two Cubes (x + y)(x2 −xy + y2 ) = x3 + y3
Difference of Two Cubes (x − y)(x2 −xy + y2 ) = x3 − y3
Sequences: Math 10 Q1 Reviewer 5
