Question

if a ,b,c are in AP as well As in GP prove that a=b=c

Answer 1

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Answer 2

a, b, c are in both AP and GP as we can see the equality of a, b and c as a = b = c  

Solution:

As the question is given that a, b, c is in both A.P. and G.P.  

To find value of b in A.P. we use $b=\frac{a+c}{2}$  

To find value of b in G.P. we use $b^{2}=a c$  

Put $b=\frac{a+c}{2}$ in $b^{2}=a c$ we get $\left(\frac{a+c}{2}\right)^{2}=a c$

$\begin{array}{l}{\frac{a^{2}+c^{2}+2 a c}{4}=a c} \\ \\{a^{2}+c^{2}+2 a c=4 a c} \\ \\{a^{2}+c^{2}-2 a c=0} \\ \\ {(a-c)^{2}=0} \\ \\{a=c}\end{array}$

Putting a in c in equation $\bold{b^{2}=a c}$ we get $\bold{b^{2}=c^{2}}$ ; b=c