Sir\Madam please explain me these questions on Arithmetic Progressions!!

We cannot entertain more than 5 questions in a subject per day. In case of multiple questions within a query, please post each question individually and let us know where you are getting stuck so that we would be able to explain things better.

Answer of question 1:

Let the two numbers be x and y.

Sum of two numbers = x + y = 13/6

Let n number of terms are inserted between x and y.

Then the total number of terms of the Ap become (n+2).

According to the given condition,

Sum of the terms of the new AP = (n + 2) + 1

⇒ (n+2)/2 {first term + last term} = (n+ 3)

⇒ ((n+2)/2) (x+y) = (n+3)

⇒ ((n+2)/2) (13/6) = (n+3)

⇒ 13n + 26 = 12n + 36

⇒ n = 10

So, 10 numbers are inserted between the given two numbers x and y.