Reviewer Math10(1st quarter)
J.Gripon
MATHEMATICS 10
SEQUENCE | chain of numbers that follow a particular pattern (its
domain (specific inputs) is finite or infinite set)
Finite sequence – list of numbers with first and last term (A1, A2,…, An)
Example:
1) 1,4,9,16,25
Pattern:
Infinite sequence – list of numbers that never ends (A1, A2,…, An,…)
Terms – individual elements in a sequence (any number in a sequence)
Rule: squaring counting numbers from one to five
Equation:
A1, A2, A3, A4, A5 - (terms of a sequence) where the subscript is the
specific number of a sequence
2) -5,-2,1,4
Pattern:
FINDING THE TERMS OF A SEQUENCE
To find the terms, substitute

( )
( )
( )
( )
( )
( )
( )
( )
( )
in the formula
Example:
Rule: the succeeding term is eight less than the product of
three and the first four counting numbers
Equation:
1. Write the first four terms of the sequence
Sol’n;
ARITHMETIC SEQUENCE |sequence of numbers to which each term
after the first term is obtained by adding a fixed number called the
common difference to the preceding term.
o
Therefore the first four terms are -1,0,1,2.
2. Finding the first three terms and the 10th term of the sequence
Where: A1 = the first term and d=common difference


Sol’n;
The nth term of an arithmetic sequence is given by:
(
)
To find the common difference d, choose any term beyond the
first and subtract the preceding term from it.
Note: sequence is arithmetic if and only if, it has a common difference
FINDING THE MISSING TERM(S) IN AN ARITHMETIC SEQUENCE
o
Therefore the first three terms are 0,2,6 and the tenth term is 90.
DERIVING A RULE FOR GENERATING A SEQUENCE

To find the rule that described the nth root term, look for the
pattern and express it as an equation.
1) Find the thirtieth term in the arithmetic sequence -4,-1,2,…
Given:
( )
Rq’d:
Sol’n:
(
)
( )
Reviewer Math10(1st quarter)
2) How many terms are there in an arithmetic sequence whose
common difference is 2 and whose first and last terms are -1
and 23 respectively?
Given:
J.Gripon
1) Insert three arithmetic means between 6 and 34.
Sol’n:
Finding d:
Solving for 3 AM:
Rq’d:
Sol’n:
(
(
)
)
Therefore, 13,20,27 are the arithmetic means of 6 and 34
2) Insert two arithmetic means between x and y.
Sol’n:
Finding d:
(
)
ARITHMETIC MEANS |term(s) between two given terms of an
arithmetic sequence
Solving for 3 AM:
Therefore,
and y.
Example: 18,38,58,78,98 where the 38,58,and 78 are the arithmetic
means of 18 and 98

Single arithmetic mean inserted between two numbers is the
arithmetic mean or the average of the two numbers. Thus, if A
is the arithmetic mean between b and c, then the arithmetic
progression is b,A,c. By definition;
are the arithmetic means of x
PARTIAL SUM OF A SEQUENCE

For the sequence A1, A2, A3, A4,…, An,… the partial sum are
Solving:

Therefore, the arithmetic mean of b and c is the average of b and c.
Example: find the arithmetic mean of 5 and 25
Sol’n:
o
Therefore, the arithmetic mean of 5 and 25 is 15.
FINDING THE ARTHMETIC MEANS OF GIVEN ARITHMETIC SEQUENCE

(formula)
Examples;
Solve for the common difference, then add it to the previous
term
and the infinite
series is
where the S1 is called first partial sum and S2 is the second
partial sum and so on.