Let two numbers are x any y then x, a, b,c,y are in A.P.
Let common difference of this A.P. = d
Then we can take
x = b-2d
a = b-d
b = b
y = b + 2d
now, a + b + c = 15
⇒ (b-d) + b + (b + d) = 15
⇒ b = 5
x + y = (b - 2d) + (b + 2d)
x + y = 2b
x+ y = 10 ......(i)
Now, x, p, q, r, y are in H.P.
Then
,,,, will be in A.P.
Let common difference of this A.P. = d'
Then we can take
=
−2d′ =
−d′ =
=
+d′ =
+2d′ Now,
++ =
⇒
=
⇒
=
and
+ =
and
+ =
=
⇒
=
(x+y=10) xy = 9 ... (ii)
x - y =
√(x2+y2)−4xy =
√(10)2−4.9 𝑥 − 𝑦 = 8 . . . . . . (𝑖𝑖𝑖)
from (i) and (iii)
x = 9
y = 1