If a, b, c are in A.P., prove that: a^2+c^2+4ac=2(ab+bc+ca)

Question

If a, b, c are in A.P., prove that:
$\text{a}^2+\text{c}^2+4\text{ac}=2(\text{ab}+\text{bc}+\text{ca})$

✓

Answer

If $\text{a}^2+\text{c}^2+4\text{ac}=2(\text{ab}+\text{bc}+\text{ca})$
Then,
$\text{a}^2+\text{c}^2+2\text{ac}-2\text{ab}=2(\text{ab}+\text{bc}+\text{ca})$
or $(\text{a}+\text{b}+-\text{c})^2-\text{b}^2=0$ $[\therefore(\text{a}+\text{b}+\text{c})^2=\text{a}^2+\text{b}^2+\text{c}^2+2\text{ab}+2\text{ac}+2\text{bc}]$
or $\text{b}=\text{a}+\text{c}-\text{b}$
or $2\text{b}=\text{a}+\text{c}$
$\text{b}=\frac{\text{a}+\text{b}}{2}$
and since,
$\text{a},\text{b},\text{c}$ are in A.P
$\text{b}=\frac{\text{a}+\text{c}}{2}$
Thus, $\text{a}^2+\text{c}^2+4\text{ac}=2(\text{ab}+\text{bc}+\text{ca})$
Hence proved.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from Arithmetic Progressions→Arithmetic Progressions — all question types→Maths STD 11 Science question papers→CBSE Board STD 11 Science question papers→CBSE Board question paper generator→

Similar questions

Prove the following by the principle of mathematical induction:
$\text{a}+\text{ar}+\text{ar}^2+...+\text{ar}^\text{n-1}=\text{a}\Big(\frac{\text{r}^\text{n}-1}{\text{r}-1}\Big),\text{r}\neq1$

→

Prove the following by the principle of mathematical induction:
$1 + 2 + 3 + ... + \text{n}=\frac{\text{n}(\text{n}+1)}{2}$ i.e, the sum of the first n natural numbers is $\frac{\text{n}(\text{n}+1)}{2}.$

→

Match the proposed probability under Column $C_1$ with the appropriate written description under column $C_2:$

	$C_1$		$C_2$
	Probability		Written Description.
$a.$	$0.95$	$i.$	An incorrect assignment.
$b.$	$0.02$	$ii.$	No chance of happening.
$c.$	$-0.3$	$iii.$	As much chance of happening as not.
$d.$	$0.5$	$iv.$	Very likely to happen.
$e.$	$0$	$v.$	Very little chance of happening.