Question

Find the common difference of the Arithmetic progression

`1/a, (3 - a)/(3a), (3 - 2a)/(3a),"......"(a ≠ 0)`

Answer 1

Given: The AP: `1/a, (3 - a)/(3a), (3 - 2a)/(3a),"......"(a ≠ 0)`. Common difference d = a₂ – a₁ by definition of AP.

Step-wise calculation:

1. `T_1 = 1/a`.

2. `T_2 = (3 - a)/(3a)`

d = T₂ − T₁

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

= `(3 - a)/(3a) - 3/(3a)`

= `(3 - a - 3)/(3a)`

= `(-a)/(3a)`

= `-1/3`

3. Verify with T₃:

`T_3 = (3 - 2a)/(3a)` 

`T_3 − T_2 = ((3 - 2a) - (3 - a))/(3a)` 

= `(-a)/(3a)`

= `-1/3`

The common difference is d = `-1/3`.