Question

the sum of two means of an ap is 44 the product of extreme is 340the what is the third term of the ap

Answer 1

Let the AP be a, a+d, a+2d, a+3d and so on, then

4a + 6d = 44 which gives

a = (22–3d)/2 = 11 -1.5d

Assume that the AP has n terms, then nth term is given as :

a + (n-1)d = 11 - 1.5d + (n-1)d = 11 - (n-2.5)d

The second equation then becomes:

(11 - 1.5d)(n-2.5d) = 140

which simplifies to

42n -110d - 6nd + 15d^2 = 560 or put simply

6n = 15d -5 - (5 * 7 * 17)/(d-7)

As n is an integer, so must be 6n and therefore so must be (5*7*17)/(d-7) which means d = 12, 14 or 24

which gives 6n = 56, 120 and 320 respectively.

Again as n is an integer, the only possible answer is n = 20, which occurs when d = 14 and a is -10 which gives us the series -10, 4, 18, 32 .. 256