Question

if a,b,c,d are in a.p and a,c,d are in g.p then show that a^2-d^2=3(b^2-ad)​

Answer 1

Step-by-step explanation:

Given that

a,b,c,d are in A.P.

We know that condition for A.P.

b-a=c-b=d-c

b=(a+c)/2  ,  c=(b+d)/2  

We can say that a+c = 2b   ---1

b+d =2 c ----2

Also given that a,c,d are in G.P.

The condition for G.P.    

$c^2=ad$   ------3

From equation 1 and 2

a+d= b+c ---4

a-d=3(b-c) ---5

From equation 4 and 5

(a-d)(a+d)=3(b+c)(b-c)

$a^2-d^2=3\left ( b^2-c^2 \right )$

Now from equation 3

$a^2-d^2=3\left ( b^2-ad \right )$

Hence prove that

$a^2-d^2=3\left ( b^2-ad \right )$