Question

Find the common difference of the AP
1/a , 3-a/3a , 3-2a/3a

Answer 1

Given,

AP = 1/a, 3-a/3a, 3-2a/3a

To find,

The common difference of the AP.

Solution,

The common difference of the AP 1/a, 3-a/3a, 3-2a/3a will be -1/3.

We can easily solve this problem by following the given steps.

We know that the difference between the two consecutive terms of an AP is always the same. So, the common difference, denoted by d, can be easily found by subtracting the first term from the second term or the second term from the third term.

In this case, the first term is 1/a and the second term is (3-a)/3a.

Common difference(d) = Second term (a2) - first term (a1)

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

d = (3-a) - (1×3)/3a [ Taking the LCM of 3a and a and adding the two fractions.]

d = 3-a-3/3a

d = -a/3a

Dividing the a in the numerator by the a in tbe denominator, we get

d = -1/3

Hence, the common difference of the AP 1/a, 3-a/3a, 3-2a/3a is -1/3.

Answer 2

The common difference of the Arithmetic Progression (A.P.) is -1/3  of 1/a , (3 - a)/3a , (3 - 2a)/3a , ...

Given:

• Arithmetic Progression (A.P.)
• 1/a , (3 - a)/3a , (3 - 2a)/3a , ...
• a ≠ 0

To Find:

• The Common difference of the Arithmetic Progression (A.P.)

Solution:

• Arithmetic sequence
• Sequence of terms in which difference between one term and the next is a constant.
• This is also called Arithmetic Progression AP
• Arithmetic sequence can be represented in the form :
• a, a + d  , a + 2d , …………………………, a + (n-1)d
• a = First term
• d = common difference = aₙ-aₙ₋₁
• nth term =  aₙ =  a + (n-1)d
• Sₙ = (n/2)(2a + (n - 1)d)

• Sum of Arithmetic sequence (AP) is called Arithmetic series

1/a , (3 - a)/(3a) , (3 - 2a)/(3a) , ...

Step 1:

Rewrite the term with common denominator 3a

3/(3a) , (3 - a)/(3a) , (3 - 2a)/(3a) , ...

Step 2:

Find Difference between consecutive terms

 (3 - a)/(3a)  - 3/(3a)  = (3 - a - 3)/(3a)  = -a/(3a)  = -1/3

 (3 - 2a)/(3a) -  (3 - a)/(3a)  = -1/3

Hence The common difference of the Arithmetic Progression (A.P.) is -1/3