Can the common difference of an AP be negative?

Progressions are numbers arranged in a particular order such that they form a predictable order. It means that from that series it can be predicted what the next numbers of that series or sequence are going to be. One of the types of progressions is arithmetic progression. Arithmetic Progression is the sequence of the numbers where the difference between any of the two consecutive numbers is the same throughout the sequence. It can also be the common difference of that series. 

Arithmetic Sequence

An Arithmetic sequence is a sequence in which every term is created by adding or subtracting a definite number to the preceding number. The definite number in the series is called a Common difference, it is denoted as “d”. Some examples of arithmetic sequence are,

• 1, 2, 3, 4, 5, 6,…
• 2, 2, 2, 2, 2, 2,…
• 22,19,16,13,10

The arithmetic sequence can be of two types, Finite sequence, and infinite sequence. Finite Arithmetic Sequence: 2, 4, 6, 8. Basically, a finite sequence has the finite number of terms. Infinite Arithmetic Sequence 2,4,6,8,10,… The infinite sequence has infinite number of terms, the three dots in the sequence represent that the sequence shall go up to infinity.

The common difference can be negative

Yes, the common difference of an arithmetic sequence can be negative. Lets first learn what is a common difference, a common difference is a difference between two consecutive numbers in the arithmetic sequence.  It is simply calculated by taking the difference between the second term and the first term in the arithmetic sequence or the difference between the third term and the second term or any of the two consecutive numbers in the sequence. Common Differences can be positive as well as it can be negative.

Example for positive common difference: 

Sequence ⇢ 2, 4, 6, 8, 10, 12,…

Common Difference = Second Term – First Term

 = 4 – 2

 = 2

The common difference of the sequence 2, 4, 6, 8, 10, 12,… is 2

Example for negative Common Difference 

Sequence ⇢ 10, 7, 4, 1, -2,…

Common Difference = Second Term – First Term

= 7 – 10

= -3

Here the common difference of the sequence 10, 7, 4, 1, -2,… is -3

Note: The common difference changes with the change of the sequence or series

Sample Problems

Question 1: Find the common difference for -1, -2, -3, -4, -5,…

Solution:

Common Difference = Second Term – First Term

= -2 – ( -1 )

= -2 + 1

= -1

The common difference of the sequence -1, -2, -3, -4, -5,… is -1

Question 2: Find the common difference for the sequence, 100, 90, 80, 70, 60,…

Solution:

Common Difference = Second Term – First Term

= 90 – 100

= -10

The common difference of the sequence 100, 90, 80, 70, 60,... is -10

Question 3: Find the 15^th term of the sequence 9, 7, 5, 3, 1,…

Solution:

a = 9
d = 7 – 9 = 5 – 7 = -2
n = 15
a₁₅ = a + ( n – 1 ) × d
= 9 + ( 15 – 1 ) × (-2)
= 9 + ( 14 ) × ( -2 )
= 9 – 28
a₁₅ = -21
Hence, 15^th term in the sequence 9,7,5,3,1,… is -21

Question 4: Find the 50^th term of the sequence 293, 290, 287, 284, 281,…

Solution:

a = 293

d = 290 – 293 = 287 – 290 = -3

n = 50

a₅₀  = a + (n – 1) × d

= 293 + ( 50 – 1 ) × (-3)

= 293 + ( 49 ) × ( -3 )

= 293 – 147

a₅₀ = 146

Hence, 50^th term in the sequence 293, 290, 287, 284, 281,… is 146

Question 5: Find the 100^th term of the sequence 100, 50, 0, -50, -100,…

Solution:

a = 100

d = 50 – 100 = 0 – 50 = -50

n = 100

a₁₀₀  = a + ( n – 1 ) × d

= 100 + ( 100 – 1 ) × (-50)

= 100 + ( 99 ) × ( -50 )

= 100 – 4950

a₅₀ = -4850

Hence, 50th term in the sequence 100, 50, 0, -50, -100,... is -4850