sequences and series

Three numbers form an increasing GP. If the middle term is doubled, then the new numbers are in AP. The commonratio of the GP will be

• $2 - \sqrt{3}$

• $2 \pm \sqrt{3}$

• $3\sqrt{2}$

• $2 + \sqrt{3}$

If in an AP, 3rd term is 18 and 7th term is 30, the sum of its 17 terms is

• 600

• 612

• 624

• None of these

The sum of the digitsin the unit place of all the numbers formed with the help of 3, 4, 5, 6 taken all at a time is

• 432

• 108

• 36

• 18

If a, b, c are in HP, then the value of $\frac{\mathrm{b} +\hspace{0.17em}\mathrm{a}}{\mathrm{b} - \mathrm{a}} + \frac{\mathrm{b} + \mathrm{c}}{\mathrm{b} - \mathrm{c}}$ is

• 0

• 1

• 2

• 3

If the sides of a $∆$ABC are in AP and a is the smallest side, then cos(A) equals

• $\frac{3\mathrm{c} - 4\mathrm{b}}{2\mathrm{c}}$

• $\frac{3\mathrm{c} - 4\mathrm{b}}{2\mathrm{b}}$

• $\frac{4\mathrm{c} - 3\mathrm{b}}{2\mathrm{c}}$

• $\frac{4\mathrm{c} - 3\mathrm{b}}{2\mathrm{b}}$

If ${\log }_{\mathrm{x}}\left(\mathrm{a}\right), {\mathrm{a}}^{\raisebox{1ex}{$\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} \mathrm{and} {\log }_{\mathrm{b}}\left(\mathrm{x}\right)$ are in GP, then x is equal to

• ${\log }_{\mathrm{b}}\left({\log }_{\mathrm{a}}\left(\mathrm{b}\right)\right)$

• $\log \left(\log \left(\mathrm{a}\right)\right)$

• $\log \left(\log \left(\mathrm{a}\right)\right) + \log \left(\log \left(\mathrm{b}\right)\right)$

• ${\log }_{\mathrm{a}}\left({\log }_{\mathrm{b}}\left(\mathrm{a}\right)\right)$

At the foot of a mountain the elevation of its summit is 45°. After ascending 2 km towards the mountain up an incline of 30°, the elevation changes to 60°. The height ofmountain will be

• $\sqrt{3} - 1$

• $1 - \sqrt{3}$

• $1 + \sqrt{3}$

• $\sqrt{3}$

If $\frac{\mathrm{a} + \mathrm{b}}{1 - \mathrm{ab}}, \frac{\mathrm{b} + \mathrm{c}}{1 - \mathrm{bc}}$ are in AP, then $\mathrm{a}, \frac{1}{\mathrm{b}}$ and c are in

• AP

• GP

• HP

• None of these

The number of terms in the series 105 + 103 + 101 + ... + 49 + 47 is

• 28

• 30

• 25

• 22

The geometric mean of roots of the equation x² - 18x + 81 = 0 is

• 18

• 6

• 9

• 3