
UNIT
10 :
SEQUENCES AND SERIES
LESSON 1:
ARITHMETIC SEQUENCES
Arithmetic
Sequences:
A
sequence such as –2, 3, 8, 13, … is called an Arithmetic
Sequence. These sequences have the following properties.
·
Terms
are denoted as t_{1} , t_{2} , t_{3 , }referring
_{ }to term1, term 2,
term 3 …
·
The difference between successive terms is constant. ie t_{2}
– t_{1} = t_{3} – t_{2} = t_{4} – t_{3}
etc
·
This
difference is called the common
difference and denoted
using the letter d. Here d
= 5.
·
The
first term is denoted using the letter a. Here a = 2.
·
Successive
terms are found by adding the common difference, d, to the preceding
term. Hence t_{5} = 13 +
5 = 18 etc.
·
The
formula for the general term or nth term is t_{n }= a + (n
– 1)d
·
Arithmetic
sequences are linear functions with domain the natural numbers N =
{1, 2, 3, 4, 5, …}
Example 1:
For the
arithmetic sequence above, find t_{7},
t_{11} and the general term t_{n}.
Solution:
a = 2 d = 5 n = 7, 11, n
Example 2: Finding specific
terms and the General (nth) Term.
Given
the sequence –3, 7, 11, 15, …
a) Show
that the sequence is arithmetic.
b)
Find t_{7}, t_{11} and
the general term t_{n}.
Solution:
a =  3 d =  4 n = 7, 11, n
Example 3: Finding the
Number of Terms in a Given Sequence.
Given
the sequence 4, 1, 2, …, 65. Find the
number of terms in the sequence.
Solution:
Let the
last term be t_{n}.
a = 4 d =  3 n = ? t_{n} =  65
Example 4: Solving a
Sequence given two terms.
The fourth and seventh terms of an arithmetic sequence are 8 and 17 respectively. Find a, d and t_{n}
Solution: