Question

If a,b,c,d,e are in AP prove a+e=b+d=2c

Answer 1

let the first term be A and common difference D,
a= A
b=A+D
c= A+2D
d= A+3D
e= A +4D
so,a+e = A+A+4D
= 2A+4D
=2(A+2D)
= 2c
and,b+d= A+D+A+3D
= 2A+4D
= 2(A+2D)
= 2c ,,proved

Answer 2

We have,
a,b,c,d,e are in AP 

To prove: a+e = b+d = 2c

Proof: Let the first term of AP be a and common difference be d
∴ b = a+d.....(1)
   c = a +2d....(2)
   d = a+3d....(3)
   e = a+4d...(4)

∴ a+e = a + a+4d = 2a + 4d = 2(a+2d) = 2c  and,    ] ................. (from 2)
   b+d = a +d + a + 3d = 2a + 4d = 2( a+2d) = 2c     ]

Hence Proved