An arithmetic sequence is a sequence where every term after the first is obtained by adding a constant called the common difference.
The sequences 1,4,7,10,... and 15, 11, 7, 3,... are examples of arithmetic sequences since each one has a common difference of 3 and -4.
Arithmetic Rule
an= a1+(n - 1)d•a1 is the first term in the sequence•n is the number of the term you are trying to determine•d is the common difference•an is the value of the term that are looking for
Try this…Use the arithmetic
formula to determine the 100th term of the following sequence:
75, 25, -25, -75, -125,…
an = a1 + (n - 1)d = 75 + (100 – 1)(-50)
= -4875a1 = 75n = 100d = -50
Try this…Use the arithmetic
formula to determine the 10th term of the following sequence:
5, 12, 19, 26,…
an = a1 + (n - 1)d = 5 + (10 – 1)(7)
= 68a1 = 5n = 10d = 7
What is missing?
1. 3, 12, 21, __ , __ , __30 39 48
2. 8, 3, -2, __ , __-7 -12
3. 5, 12, __ , 26, __ 19 33
What is missing?
4. __, __,__, 8 , 12 , 16-4 0 4
5. -1, __ , __,__, 31, 39 7 15
6. 13,__ ,__,__,-11.-17 7 123
-5
Insert three terms between 2 and 34 of an
arithmetic sequence.
ARITHMETIC MEANS
The terms between any two consecutive terms of an arithmetic sequence.
Insert 4 arithmetic means between 5 and
25.
Solution: Since we are required to
insert 4 terms, then there will be 6 terms in all.
Let and 5 ,𝑎2 ,𝑎3 ,𝑎4 ,𝑎5 ,25
Let 5 ,𝑎2 ,𝑎3 ,𝑎4 ,𝑎5 ,25
25=5+ (5 ) 𝑑20−5=5𝑑4=𝑑
5 ,𝑎2 ,𝑎3 ,𝑎4 ,𝑎5 ,25𝑎2=𝑎1+(1)(𝑑 ) n = 2
(2 – 1) = 1𝑎2=5+(1 ) (4 )=9=13
𝑎4=5+ (3 ) (4 )=17𝑎5=5+(4 ) (4 )=21
Insert 2 arithmetic means between ½ and 2.
ARITHMETIC SUMSum of the terms in
an arithmetic sequence.
What is 1 + 2 + 3 + …+ 50 +
51 +… + 98 + 99 +100?
Formula: 𝑎𝑛=𝑎1+ (𝑛−1 ) 𝑑
Example:
Find the sum of the first 10 terms of the arithmetic sequence 5, 9, 13, 17,…
Using the formula:
𝑆10=102
[2(5)+(10−1 ) 4]
d = 9 - 5
𝑆10=5 [10+ (9 )4 ]=5 (46 )=𝟐𝟑𝟎
Example:
Find the sum of the first 20 terms of the arithmetic