Question
If $a, b, c, d$ are in continued proportion, prove that:
$(a^2 + b^2 + c^2) (b^2 + c^2 + d^2) = (ab + bc + cd)^2.$
$(a^2 + b^2 + c^2) (b^2 + c^2 + d^2) = (ab + bc + cd)^2.$
✓
Answer
$L.H.S.$
$= (d^2k^6 + d^2k^4 + d^2k^2)(d^2k^4 + d^2k^2 + d^2)$
$= d^2k^2(k^4+ k^2 + 1)d^2 (k^4 + k^2 + 1)$
$= d^4k^2 (k^4 + k^2 + 1)^2$
$R.H.S.$
$= (ab + bc + cd)^2$
$= (dk^3·dk^2 + dk^{2·}dk + dk·d)^2$
$= d^4·k^2 (k^4 + k^2+ 1)^2$
$L.H.S. = R.H.S.$
Hence proved.
$= (d^2k^6 + d^2k^4 + d^2k^2)(d^2k^4 + d^2k^2 + d^2)$
$= d^2k^2(k^4+ k^2 + 1)d^2 (k^4 + k^2 + 1)$
$= d^4k^2 (k^4 + k^2 + 1)^2$
$R.H.S.$
$= (ab + bc + cd)^2$
$= (dk^3·dk^2 + dk^{2·}dk + dk·d)^2$
$= d^4·k^2 (k^4 + k^2+ 1)^2$
$L.H.S. = R.H.S.$
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.
Explore more
Similar questions
In the following figure, DE || Ac and DC || AP. Prove that:$\frac{B E}{E C}=\frac{B C}{C P}$
→Prove the following identity:
$
(1-\tan A)^2+(1+\tan A)^2=2 \sec ^2 A
$
→$
(1-\tan A)^2+(1+\tan A)^2=2 \sec ^2 A
$
In a T.T. match between Geeta and Ritu, the probability of the winning of Ritu is 0.73. Find the probability of: winning of Geeta
A bag contains 16 colored balls. Six are green, 7 are red and 3 are white. A ball is chosen, without looking into the bag. Find the probability that the ball chosen is : not red
Solve x – 3 (2 + x) > 2 (3x – 1), x ∈ { – 3, – 2, – 1, 0, 1, 2, 3}. Also represent its solution on the number line.
If $x: y = 4: 7,$ find the value of $(3x + 2y): (5x + y).$
→Which of the following lists of numbers form an A.P.? If they form an A.P., find the common difference d and write the next three terms : a, 2a, 3a, 4a,...
Given $A=\left[\begin{array}{cc}1 & 1 \\ -2 & 0\end{array}\right]$ and $B=\left[\begin{array}{cc}2 & -1 \\ 1 & 1\end{array}\right]$ Solve for matrix $X: X + 2A = B$
→In each of the following find the values of k of which the given value is a solution of the given equation:
$k x^2+\sqrt{2} x-4=0 ; x=\sqrt{2}$
→$k x^2+\sqrt{2} x-4=0 ; x=\sqrt{2}$
What number should be subtracted from $x^2 + x + 1$ so that the resulting polynomial is exactly divisible by $(x-2)$ ?
→