Question
Show that $( a - b )^2,\left( a ^2+ b ^2\right)$ and $( a + b )^2$ are in $A.P$?
✓
Answer
$(a-b)^2,\left(a^2+b^2\right)$ and $(a+b)^2$ are in A.P.
If $2\left(a^2+b^2\right)=(a-b)^2+(a+b)^2$
If $2\left(a^2+b^2\right)=a^2+b^2-2 a b+a^2+b^2+2 a b$
If $2\left(a^2+b^2\right)=2 a^2+2 b^2=2\left(a^2+b^2\right)$
Which is true
Hence proved.
If $2\left(a^2+b^2\right)=(a-b)^2+(a+b)^2$
If $2\left(a^2+b^2\right)=a^2+b^2-2 a b+a^2+b^2+2 a b$
If $2\left(a^2+b^2\right)=2 a^2+2 b^2=2\left(a^2+b^2\right)$
Which is true
Hence proved.
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