Question

Is it possible that three different numbers a, b, c may be both in A.P. and G.P. Give reasons for
your answer.​

Answer 1

Answer:

Only when all the numbers are equal.

Step-by-step explanation:

If they are in GP, b² = ac ...(1)

If they are in AP, 2b = a + c

Square on both sides,

=> (2b)² = (a + c)²

=> 4b² = a² + c² + 2ac

=> 4(ac) = a² + c² + 2ac {from (1)}

=> 0 = a² + c² + 2ac - 4ac

=> 0 = (a - c)²

=> a = c, when a & c are equal.

Thus, 2b = a + c = a + a

2b = 2a => b = a.

Hence, a = b = c.

It says, it is possible, when all numbers are equal.

Note : you can't apply the formulae for sum of finite terms of GP(since, common ratio a/b = a/a is 1)