Suppose $a,\ b,\ c$ are in AP, then prove that $b=\frac{a+c}{2}$.

Academic Mathematics NCERT Class 10

Given: $a,\ b,\ c$ are in A.P.
To do: To prove that $b=\frac{a+c}{2}$.
Solution:
$\because a,\ b,\ c$ are in A.P.
$\Rightarrow b-a=c-b$
$\Rightarrow b+b=a+c$
$\Rightarrow 2b=a+c$
$\Rightarrow b=\frac{a+c}{2}$
Hence proved.

Tutorialspoint
Simply Easy Learning

Updated on: 10-Oct-2022

21 Views

Related Articles
Prove that $ \sin \left(C+\frac{A+B}{2}\right)=\sin \frac{A+B}{2} $

If $2^{a}=3^{b}=6^{c}$ then show that $ c=\frac{a b}{a+b} $

If the angles $A,\ B,\ C$ of a $\vartriangle ABC$ are in A.P., then prove that $b^2=a^2+c^2-ac$.

If $ A, B, C $, are the interior angles of a triangle $ A B C $, prove that$ \tan \left(\frac{C+A}{2}\right)=\cot \frac{B}{2} $

If $ A, B, C $, are the interior angles of a triangle $ A B C $, prove that$ \sin \left(\frac{B+C}{2}\right)=\cos \frac{A}{2} $

If the roots of the equation $a(b-c) x^2+b(c-a) x+c(a-b) =0$ are equal, then prove that $b(a+c) =2ac$.

In $\vartriangle ABC$, if $cot \frac{A}{2},\ cot \frac{B}{2},\ cot \frac{C}{2}$ are in A.P. Then show that $a,\ b,\ c$ are in A.P.

Prove that:$ \left(\frac{x^{a}}{x^{b}}\right)^{a^{2}+a b+b^{2}} \times\left(\frac{x^{b}}{x^{c}}\right)^{b^{2}+b c+c^{2}} \times\left(\frac{x^{c}}{x^{a}}\right)^{c^{2}+c a+a^{2}}=1 $

Prove that:$ \left(\frac{x^{a}}{x^{-b}}\right)^{a^{2}-a b+b^{2}} \times\left(\frac{x^{b}}{x^{-c}}\right)^{b^{2}-b c+c^{2}} \times\left(\frac{x^{c}}{x^{-a}}\right)^{c^{2}-c a+a^{2}}=1 $

Prove that $ (a-b)^{2}, a^{2}+b^{2} $ and $ (a+b)^{2} $ are three consecutive terms of an AP.

If $A, B$ and $C$ are interior angles of a triangle $ABC$, then show that: $sin\ (\frac{B+C}{2}) = cos\ \frac{A}{2}$

Given that $sin\ \theta = \frac{a}{b}$, then $cos\ \theta$ is equal to(A) $ \frac{b}{\sqrt{b^{2}-a^{2}}} $(B) $ \frac{b}{a} $(C) $ \frac{\sqrt{b^{2}-a^{2}}}{b} $(D) $ \frac{a}{\sqrt{b^{2}-a^{2}}} $

If the roots of the equation $(b-c)x^2+(c-a)x+(a-b)=0$ are equal, then prove that $2b=a+c$.

Prove that$ \frac{a+b+c}{a^{-1} b^{-1}+b^{-1} c^{-1}+c^{-1} a^{-1}}=a b c $

If $ A, B, C $ are the interior angles of a $ \triangle A B C $, show that:$ \tan \frac{B+C}{2}=\cot \frac{A}{2} $

Kickstart Your Career

Get certified by completing the course
Get Started

Advertisements