If a,b,c are in AP as well as in GP then show that a=b=c - Maths - Sequences and Series

Question

If a,b,c are in AP as well as in GP then show that a=b=c

Aakash Sharma · Accepted Answer

Given : a, b, c are in A P
â‡’ 2b = a + c (1)
â‡’ (2b)2 = (a + c)2
â‡’ 4b2 = a2 +
c2 + 2ac (2)
Also,
a, b, c are in G P
â‡’ b2 = ac (3)
Fron (2) and (3) we get
4ac = a2 + c2 + 2ac
â‡’ a2 + c2 + 2ac - 4ac
= 0
â‡’ a2 + c2 - 2ac =
0
â‡’ (a - c)2 = 0
â‡’ a - c = 0
â‡’ a = c (4)
From (1) and (4) we get
2b = a + a = c + c
â‡’ 2b = 2a = 2c
â‡’ a = b = c

If a,b,c are in AP as well as in GP then show that a=b=c.