Question

Show that each of the progressions given below is an AP. Find the first term, common difference and next term of each.
(v) `sqrt(20)`, `sqrt(45)`, `sqrt(80)`, `sqrt(125)`,.........

Answer 1

This given progression `sqrt(20)`, `sqrt(45)`, `sqrt(80)`, `sqrt(125)`,......... 
This sequence can be re-written as 2 `sqrt(5),3 sqrt(5),4 sqrt(5), 5 sqrt(5),`..........
Clearly, `3 sqrt(5) -2 sqrt(5) = 4 sqrt(5) -3  sqrt(5) =5  -4 sqrt(5) = sqrt(5)` (Constant)
Thus, each term differs from its preceding term by `sqrt(5)`.So, the given progression is an AP.
First term = 2 `sqrt(5)` = `sqrt(20)`
Common difference = `sqrt(5)`
Next term of the AP =  5 `sqrt(5)  + sqrt(5) =  6sqrt(5) = sqrt(180)`