Question

write the condition under which three numbers a, b, c may be in A.P. and G.P. both

Answer 1

Condition for being in AP

is b = a+c/2

Answer 2

The condition is $\bold{a = b = c}$

Step-by-step explanation:

The condition for any three numbers to be in A.P is that the common difference should be same between any two consecutive numbers.

i.e., for a, b, c to be in A.P  

$b - a = c - b$

$2 b=a+c \rightarrow(1)$

The condition for any three numbers to be in G.P is that the common ratios should be same between any two consecutive numbers.

i.e., for a, b, c to be in G.P

$\frac{b}{a}=\frac{c}{b}$

$b^{2}=a c$

$a=\frac{b^{2}}{c}$

Put value of a in (1)

$2 b=\frac{b^{2}}{c+c}$

$2 b=\frac{b^{2}+c^{2}}{c}$

$b^{2}+c^{2}=2 b c$

$b^{2}+c^{2}-2 b c=0$

$(b-c)^{2}=0$

$b - c = 0$

$b = c$

Put b = c in (1), we get,

$2c = a + c \Rightarrow c = a$

Therefore, a = b = c is the condition for all the three numbers to be in A.P and G.P

a, a, a are in A.P with common difference = a - a = 0  

a, a, a are in G.P with common ratio a/a = 1