Question

If n A.M’s are inserted between two quantities a and b, then their sum is equal to?

Answer 1

`S_"(n+2)" = ((n + 2))/2 (a + b)`

Explanation:

If there are "n" Arithmetic Means inserted between "a" and "b", then we understand that there is a total of (n + 2) Terms in an Arithmetic Progression ( AP ).

In our problem,

First Term = "a"

Last Term = "b"

Total number of terms = ( n + 2 )

General formula to find the SUM of "n" terms in an Arithmetic Progression is given by

`S_"n" = ( n / 2 ) ( a_(1) + a_"n")`, where `a_(1)` is the First Term and `a_(n)` is the Last Term of the AP.

Referring to our problem situation,

Common Difference (d) = `(b-a)/(n + 1)`

Hence, our AP can be written as follows:

`a , a + [(b-a)/(n+1)], a + 2[(b-a)/(n + 1)], a + 3[(b-a)/(n+1)], .................., b`

Sum of (n + 2) terms of AP is given by

`S_"(n+2)" = ((n + 2))/2 (a + b)`