2. Arithmetic sequence
is a set of numbers listed in a specific order such that
the difference between succeeding terms is constant
is a sequence in which each term is obtained from
the preceding term by adding a common difference.
Term Each number in the list of the sequence and is
denoted by 𝑎1 , 𝑎2, …,𝑎𝑛 , where 𝑎𝑛 is the nth term
in the sequence.
3. Common difference
This is any two consecutive terms have a constant
difference of the given arithmetic sequence and
denoted by d
n 1 2 3 4 5
f (n) 30 25 20 15 10
𝑓 𝑛 = 𝑎1+ (n-1)d
the nth term
the first term a multiple that is less
than n
the common difference
4. Use the formula for Arithmetic Sequence
𝑎𝑛 = 𝑎1 + ( n - 1 ) d
Example 1: Find the 47th term given the sequence of
numbers 5, 13 , 21 , 29, 37
Solution:
Given: 𝑎𝑛 = 𝑎47 ; 𝑎1 = 5 ; n = 47 ; d = 8
𝑎𝑛 = 𝑎1 + ( n - 1 ) d
𝑎47 = 5 + ( 47 - 1 ) 8
𝑎47 = 5 + ( 46) 8
𝑎47 = 5 + 368
𝑎47 = 373
5. Use the formula
d = 𝑎𝑛 - 𝑎𝑛−1
Example 2: Find the 12th term in the sequence 11x-5 , 14x-2 , 17x +1 ,
:Solution Given: 𝑎𝑛 = 𝑎12 ; 𝑎11 = 11𝑥 − 5; n = 12 ; d = (14x -2) – (11x – 5)
Solve for the common difference
d = (14x -2) – (11x – 5)
d = 14x -2– 11x + 5
d = 14x -2– 11x + 5
d = 14x – 11x -2 + 5
d = 3x + 3
Solve for the 12th term
𝑎𝑛 = 𝑎1 + ( n - 1 ) d
𝑎12 =11x - 5 + ( 12 - 1 ) (3x + 3)
𝑎12 =11x - 5 + ( 11 ) (3x + 3)
𝑎12 =11x - 5 + (33x + 33)
𝑎12 =11x - 5 + 33x + 33
𝑎12 = 44x + 28
6. 𝑎𝑛 = 𝑎1 + ( n - 1 ) d
Example 3: In the arithmetic sequence 21, 16, 11, 6, …, which term is -34
Solution:
Given: 𝑎𝑛 = -34 ; 𝑎1 = 21 ; n = ? ; d = -5
Solve for number of terms:
𝑎𝑛 = 𝑎1 + ( n - 1 ) d
−34 = 21 + ( n - 1 ) -5
−34 = 21 -5 n + 5
−34 -21-5= -5 n
−60 = -5 n
−60
−5
=
−5𝑛
−5
𝑛 = 12 Thus, -34 is the 12th term:
7. Also known as average, is a number calculated by adding
two numbers a and b and dividing by the number of terms
in the set denoted by 𝑎 + 𝑏
2
𝑎+𝑏
2
=
15+27
2
= 21
Insert an arithmetic mean between 15 and 27
Solution 2: Solve for d
𝑎𝑛 = 𝑎1 + ( n - 1 ) d
27 = 15 + ( 3 - 1 ) d
27 -15= 2d
d = 6
Adding the common difference, 6, with
the first term, 15 is the arithmetic mean.
Thus, the arithmetic mean is 15+6 = 21
8. The sum of the terms in an arithmetic sequence given by:
𝑠𝑛 =
𝑛
2
( 𝑎1 + 𝑎𝑛)
Where 𝑠𝑛 is the sum of the terms;
𝑎1 is the first term;
𝑑 is the common difference;
n is the number of terms.
9. 𝑎𝑛 = 𝑎1 + ( n - 1 ) d
Example 4: Find the sum of the first 24 terms of an arithmetic
sequence where the first term is 7 and the common difference is 5.
Find 𝑎𝑛
𝑎24 = 7 + ( 24 - 1 ) 5
𝑎24 = 7 + ( 23 ) 5
𝑎24 = 7 + 115
𝒂𝟐𝟒 = 122
Then , find the sum.
𝑠𝑛 =
𝑛
2
( 𝑎1 + 𝑎𝑛)
𝑠24 =
24
2
( 7 + 122)
𝑠24 =
24
2
(129)
𝑠24 = 12 (129)
𝒔𝟐𝟒 = 𝟏𝟓𝟒𝟖
10. 1. Given the sequence of numbers 15 , 9 , 3 , -3 , -9 , …., find the 73rd term.
2. Find the 28th term given that 𝑎1 = -1 and d = 5
3. The seat plan of a concert venue is arranged in a sequence. The first
row has 5 sets. The seats for the succeeding row are 3 more than the
previous. If there are 42 rows, what is the seating capacity of the concert
venue?
Find the um of the arithmetic sequence
