Find three numbers in G.P. whose sum is 38 and their product is 1… - Vidyadip

Question

Find three numbers in G.P. whose sum is 38 and their product is 1728.

✓

Answer

Let the three number be $a, ar, ar^2$ in G.P., where a is first teror and r is the common ratio.Then,
$\text{a} + \text{ar} +\text{ar}^2 = 38$
$\text{a} (1 + \text{r} + \text{r}^2) = 38\cdots(\text{i})$
and
$\text{(a)}\text{(ar)}\text{(ar)}^2=1728$
$\text{a}^3\text{r}^3=1728=4^33^3=(12)^3$
$\text{a}^3=\frac{12^3}{\text{r}^3}\Rightarrow\frac{12}{\text{r}}=\text{a}$
Putting $\text{a}=\frac{12}{\text{r}}\text{ in }(\text{i})$
$\frac{12}{\text{r}}(1 + \text{r} + \text{r}^2)=38$
$12+12\text{r}+12\text{r}^2=38\text{r}$
$12\text{r}^2-26\text{r}+12=0$
$6\text{r}^2-13\text{r}+6=0$
$6\text{r}^2-9\text{r}-4\text{r}+6=0$
$3\text{r}(3\text{r}-3)-2(3\text{r}-3)=0$
$\text{r}=\frac{3}{2},\frac{2}{3}$
$\text{a}=\frac{12}{\frac{3}{2}}=8\text{ or }\frac{12}{\frac23}=18$
$\therefore$ G.P. is 8, 12, 18.

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 "(Each question 4 marks)" from Geometric Progressions→Geometric Progressions — all question types→MATHS STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

If a, b, c are in G.P., prove that: $\frac{1}{\text{a}^2-\text{b}^2}+\frac{1}{\text{b}^2}=\frac{1}{\text{b}^2-\text{c}^2}$

Out of 18 points in a plane, no three are in the same straight line except five points which are collinear. How many:

Straight lines.
Triangles can be formed by joining them?

For three sets A, B and C, show that.
$\text{A}\cap\text{B}=\text{A}\cap\text{C}$ need not imply B = C.

Solve the following systems of linear inequations graphically: $\text{x}-\text{y}\leq1,\text{x}+2\text{y}\leq8,2\text{x}+\text{y}\geq2,\text{x}\geq0$ and $\text{y}\geq0$

Solve the following equation: $\sqrt{3}\text{x}^2-\sqrt{2}\text{x}+3\sqrt{3}=0.$

Evaluate the following limit: $\lim\limits_{\text{h}\rightarrow0}\frac{\sqrt{\text{x}+\text{h}}-\sqrt{\text{x}}}{\text{h}},\text{x}\ne0$

Life of bulbs produced by two factories A and B are given below:

Length of life (in hours):	550-650	650-750	750-850	850-950	950-1050
Factory A: (Number of bulbs)	10	22	52	20	16
Factory B: (Number of bulbs)	8	60	24	16	12

The bulbs of which factory are more consistent from the point of view of length of life?

If $\text{A}\times\text{b}\subseteq\text{C}\times\text{D and A}\times\text{B}=\phi,$ prove that $\text{A}\subseteq\text{C and B}\subseteq\text{D}$

In each the following find the equation of the hyperbola satisfying the given conditions: Foci $(0, \pm13), $ conjugate axis = 24

Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow-\infty}\big(\sqrt{\text{x}^2-8\text{x}}+\text{x}\big)$