What is the common difference of the AP 1/b, (3-b)/3b, (3-2b)/3b…?

Common difference of the given AP (1/b , (3-b) / 3b , (3-2b) / 3b … ) is -1/3 and can be calculated using the general formulas for finding the common difference of an AP . In this article, we are going to learn about the concept of common differences, explore methods to determine them , and apply our knowledge to solve similar problems efficiently.

Common Difference

Common difference is the difference between two consecutive terms of an Arithmetic progression . Common difference always remains constant for an A.P. and is denoted by (d) usually .The common difference of an Arithmetic Progression can be positive, negative, or zero.

We can find the common difference of an A.P. is constant for every pair of numbers. If there are two common differences present in sequence then it is not an Arithmetic progression.

Terms of an arithmetic progression can be algebraically represented as

a, a + d, a + 2d, a + 3d, a + 4d, a + 5d…

where,

• a is First Term
• d is Common Difference

Formula for calculating the common difference in Arithmetic progression is d = a_n – a _n-1 , where a_n and a_n-1 are the n^th and (n-1) ^th term of A.P.

Steps to Calculate Common Difference

1. Take any term of the given sequence
2. Subtract the term from its succeeding(next) term
3. Difference of the terms is the common difference

For example: Let us calculate the common difference of the a arithmetic sequence 1, 3, 5, 7, 9. 

Given that the sequence is 1, 3, 5, 7, 9.

Common difference can be calculated as

d = 3 – 1 = 5 – 3 = 7 – 5 = 9 – 7

d = 2

Hence, the common difference of the given sequence is 2.

Properties of Common Difference

1. In an AP, the common difference remains constant throughout the sequence. This means that the difference between any two consecutive terms is always the same.
2. When we multiply an AP with a constant term (say k ) then the new common difference will be k times the previous common difference.
3. When two AP are added / subtracted then the common difference of the resulting AP Will be the sum / difference of the common difference of given two AP
4. When two AP are Multiplied / divided then the common difference of the resulting AP Will be the multiplication / quotient of the common difference of given two AP .
5. A positive value of common difference shows the AP is increasing AP.
6. A negative value of common difference shows the AP is decreasing AP.
7. A null value of common difference (or 0) shows the AP is constant AP.

Note that here addition operation refers to adding the 1st , 2nd , 3rd . . . . nth term of first AP in the 1st , 2nd , 3rd . . . . nth term of the second AP and same goes for subtraction, multiplication, division.

What is Common Difference of AP 1/b, (3-b)/3b, (3-2b)/3b…?

As explained above, let’s solve the AP mentioned in the problem statement:

Given that the arithmetic progression is 1/b, (3-b)/3b. (3-2b)/3b.

The common difference can be calculate as,

d = (3-b)/3b-1/b

d = (3-b-3)/3b

d = -b/3b

d = -1/3

Hence, the common difference of the AP is -1/3

Conclusion

In summary, the arithmetic progression (AP) represented by the terms 1/b, (3-b)/3b, (3-2b)/3b… is -1/3 . Understanding the common difference of an AP is important for understanding its structure and behaviour , shedding light on its mathematical properties and practical applications. By exploring common differences, we deepen our understanding of arithmetic progressions, enhancing our comprehension of mathematical concepts and their relevance in various scenarios.

Similar Problems

Problem 1: Calculate the common difference of the AP 156, 131, 106,  81, 56.

Solution:

Common difference can be calculated as

d = 131-156

d = 25

Hence, the common difference of the AP is 25.

Problem 2: If the 10th term of an AP is 50 and the 15th term is 65, find the common difference of the AP.

Solution:

Given

10^th term = a + 9d = 50

15^th term = a + 14d = 65

subtracting both terms

(a + 14d ) – ( a+9d ) = 65 – 50

5d = 15

d = 3

Hence, the common difference of the AP is 3.

Problem 3: Find the common difference of the AP, 99, 88, 77, 66…

Solution:

Common difference can be calculated as

d = 88 – 99

d = -11

Hence, the common difference of the AP is -11.

Problem 4 : The first term of an AP is 3 and the last term is 31. If there are 15 terms in the AP, find the common difference.

Solution:

Given a =3

a_n=a + (n-1 )d = 31 , where n = 15

= 3 + (15-1 ) d = 31

=> 14d = 28

=> d = 2

Hence the common difference of the given AP is 2

Problem 5 : Find the common difference of the AP : log2 , log4 , log8 , log16 , log32 …… .

Solution:

From the Given Sequence

a = log2

a + d = log4

So, (a+d ) – a = log4 – log2

d = log4/2 = log2

Hence the common difference of the AP is log2