If a, b, c, d are in continued proportion, prove that: a^3+b^3+c^… - Vidyadip

Question

If $a, b, c, d$ are in continued proportion, prove that:
$
\frac{a^3+b^3+c^3}{b^3+c^3+d^3}=\frac{a}{d}
$

✓

Answer

$a, b, c, d$ are in continued proportion
$
\begin{aligned}
& \therefore \frac{a}{b}=\frac{b}{c}=\frac{c}{d}= k \text { (say) } \\
& \therefore c = dk _{ t } b = ck = dk . k = dk ^2, \\
& a = bk = dk ^2 \cdot k = dk ^3 \\
& \text { L.H.S. }=\frac{a^3+b^3+c^3}{b^3+c^3+d^3} \\
& =\frac{(d k)^3+\left(d k^2\right)^3+(d k)^3}{\left(d k^2\right)^3+(d k)^3+d^3} \\
& =\frac{d^3 k^9+d^3 k^6+d^3 k^3}{d^3 k^6+d^3 k^3+d^3} \\
& =\frac{d^3 k^3\left(k^6+k^3+1\right)}{d^3\left(k^6+k^3+1\right)} \\
& = k ^3 \\
& \text { R.H.S. }=\frac{a}{d} \\
& =\frac{d k^3}{d} \\
& = k ^3 \\
& \therefore \text { L.H.S. }=\text { R.H.S. }
\end{aligned}
$

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 "[4 marks sum]" from Ratio and Proportion→Ratio and Proportion — all question types→Mathematics STD 10 question papers→ICSE Board STD 10 question papers→ICSE Board question paper generator→

Similar questions

The shadow of a vertical tower on a level ground increases by $10\ m$ when the altitude of the sun changes from $45^\circ$ to $30^\circ .$ Find the height of the tower, correct to two decimal places.

Solve each of the following equations using the formula:
$\frac{x-1}{x-2}+\frac{x-3}{x-4}=3 \frac{1}{3}$

The price of a T.V. set inclusive of sales tax at the rate of $14\%$ is Rs $30,552.$ Find the basic price of the T.V. set.

A circus tent is cylindrical to a height of 5 m and conical above it. If its diameter is 42 m and slant height of the cone is 53 m, calculate the total area of the canvas required.

Draw an angle ABC = 75° Draw the locus of all the points equidistant from AB and BC.

A lady holds 1800, Rs 100 shares of a company that pays 15% dividend annually. Calculate her annual dividend. If she had bought these shares at 40% premium, what is the return she gets as per cent on her investment? Give your answer to the nearest integer.

A shopkeeper buys a camera at a discount of 20% from the wholesaler, the printed price of the camera being ₹ 1600 and the rate of sales tax is 6%. The shopkeeper sells it to the buyer at the printed price and charges sales tax at the same rate. Find
(i) the price at which the camera can be bought.
(ii) the VAT (Value Added Tax) paid by the shopkeeper.

Let $A=\left[\begin{array}{cc}2 & 1 \\ 0 & -2\end{array}\right], B=\left[\begin{array}{cc}4 & 1 \\ -3 & -2\end{array}\right]$ and $C=\left[\begin{array}{ll}-3 & 2 \\ -1 & 4\end{array}\right]$. Find $A^2 + AC -5B$.

If $a, b, c$ and $d$ are in proportion, prove that:
$
\frac{a^2+a b+b^2}{a^2-a b+b^2}=\frac{c^2+c d+d^2}{c^2-c d+d^2}
$

Two circles intersect in points P and Q. A secant passing through P intersects the circles in Aand B respectively. Tangents to the circles at A and B intersect at T. Prove that A, Q, B and T lie on a circle.