1. In a geometric progression consisting of positive terms, each term equals the sum of the next two terms. The common ration of the G.P. is

2. If p and q are positive real numbers such that $p^2+q^2$=1, then the maximum value of (p + q) is

3. If $x=\sum _{n=0}^{\infty }{a}^n$, $y=\sum _{n=0}^{\infty }{b}^n$, $z=\sum _{n=0}^{\infty }{c}^n$, where a, b, c are in A.P. and $\left|a\right|<1,\left|b\right|$ $<1,\left|c\right|$ $<$1, then x, y, x are in

4. Let $T_r$ be the $r^{\mathrm{th}}$ term of an A.P. whose first term is a and common difference is d. If for some positive integers m, n, m $\ne n,T_m$=$\frac{1}{n}$, $T_n$=$\frac{1}{m}$, then a $-d$=

5. Let two numbers have A.M. 9 and G.M. 4. Then the two numbers are the roots of

6. If the fifth term of a G.P is 2, then the product of the first 9 terns is

7. The value of $2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}$ $·$ $4^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$8$}\right.}$ $·$ $8^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$16$}\right.}$ .............$\infty$is

8. In a G.P if the sum of infinite terms is 20 and the sum of their squares is 100, then the common ratio is

9. $1^3$ $-$ $2^3$+$3^3$ $-4^3$+........+$9^3$=

10. For 0$<\theta <\frac{\pi }{2}$, if $x=\sum _{n=0}^{\infty }$${\cos }^{2n}\theta {\sin }^{2n}\theta ,$then xyz=

11. An infinite G.P has first term x and sum 5. Then

12. ${11}^3-{10}^3+9^3-8^3$+$7^3$ $-6^3$ $+5^3$$-4^3+3^3-2^3+1^3$ =

13. If x and y are positive real numbers and m. n are positive integers, then the minimum value of $\frac{x^m{y}^n}{(1+x^{2m})(1+y^{2n})}$ is

14. Let the positive numbers a, b, c, d be in A.P. Then abc, abd, acd, bcd are in

15. Suppose a, b, c are in A.P., and $a^2,b^2,c^2\mathrm{are}$in G.P. If $a<b<c$and a + b + c = $\frac{3}{2}$ Then a =

16. Let $\alpha ,\beta$ be the roots of $x^2-x+p=0$ and $\gamma ,\delta$ be the roots of $x^2-4x+q=0$. If $\alpha ,\beta ,\gamma ,\delta$ are in G.P. then the integral values of p and q respectively, are

17. If the sum of the first 2n terms of the A.P. 2, 5, 8, ....... is equal to the sum of the first n terms of the A.P. 57, 59, 61, ......., then n =

18. If $x>1,y>1,z>1$ are in G.p, then
$\frac{1}{1+1nx}$ , $\frac{1}{1+1ny}$, $\frac{1}{1+1nz}$ are in

19. The fourth power of the common difference of an A.P with integeter entries is added to the product of any four consecutive terms of it. The resulting sum is

20. If a, b, c, d are positive real numbers such that a + b + c+ d = 2, then M = (a +b) (c + d) satisfies the relation

21. Consider an infinite geometric series with first term a and common ratio r. If its sum is 4 and the second term is $\frac{3}{4}$, then

22. Let $a_1$, $a_2$, ...$a_{10}$ be in A.P. and $h_1$, $h_1$, ....$h_{10}$ be in H.P. If $a_1$= $h_1$= 2 and $a_{10}$= $h_{10}$=3, then $a_4{h}_7$ is

23. Let $T_r$ be the $r^{\mathrm{th}}$ term of an A.P. for r = 1, 2, 3, .........If for some positive integers m and n we have $T_m$= $\frac{1}{n}$ and $T_n$= $\frac{1}{m}$, then $T_{\mathrm{mn}}$=

24. Let p and q be the roots of the equation $x^2-2x+A=0$ and let r and s be the roots of the equation $x^2-18x+B=0$. If $p<q<r<s$ are in A.P, then $A+B=$

25. If the produc of n positive numbers is unity, then their sum is

26. Let x be the arithmetic mean and y, z be the two geometrical means between and two positive numbers. The value of $\frac{y^3+z^3}{\mathrm{xyz}}$ is

27. If $\cos (x-y),\cos x,\cos (x+y)\mathrm{are}\mathrm{in}H\mathrm{.}P,\mathrm{then}\mathrm{the}\mathrm{value}\mathrm{of}\cos x\sec \frac{y}{2}$ is

28. If the function $f$ satisfies the relation $f(x+y)$ = $f(x)·f(y)$ for all natural numbers x, y, $f(1)$ = 2 and $\sum _{r=1}^n$ $f(a+r)$ = 16 ($2^n$ $-1)$, then the natural number $a$ is

29. Let the H.m and G.M of two positive numbers a and b be in the ratio 4 : 5 then a : b is

30. If $s_1$,$s_2$ , $s_3$..... are the sum of infinite geometirc series whose first terms are 1, 2, 3, ............... and whose common rations $\frac{1}{2}$, $\frac{1}{3}$, $\frac{1}{4}$, ......respectively, then ${s_1}^2+{s_2}^2$+${s_3}^2$+.........+${s_{10}}^2$ =

31. The sum of the first n terms of the series, $1^2$ + 2$·2^2$ + $3^2$ + 2$·4^2$ + $5^2$ + 2$·6^2$+... is $\frac{n}{2}$ ${(n+1)}^2$, when n is even. When n is odd the sum is

32. If the first and ${(2n-1)}^{\mathrm{th}}$ terms of an A.P., a G.P. and a H.P. are equal and their $n^{\mathrm{th}}$terms are a, b, c respectively, then

33. If a, b, c, d and p are distinct real numbers such that ($a^2$+$b^2$+$c^2$) $p^2$ $-$ 2 (ab + bc+ cd)p + $b^2$ + $c^2$+ $d^2$ $\le$ 0. Then a,b, c, d are

34. The sum of the first n terms of the series $\frac{1}{2}$ + $\frac{3}{4}$ + $\frac{7}{8}$+$\frac{15}{16}$+.... is

35. The sum of integers from 1 to 100 which are divisible by 2 or 5 is

36. Let a, b, c three numbers between 2 and 18 such that their sum is 25. If 2, a, b are in A.P and b, c, 18 are in G.P., then c = --------

37. If $a_1$, $a_2$, ........$a_n$ are in A.P, where $a_i$ $>$ 0 for all i, then $\frac{1}{\sqrt{a_1}+\sqrt{a_2}}$ + $\frac{1}{\sqrt{a_2}+\sqrt{a_3}}$ + .....+ $\frac{1}{\sqrt{a_{n-1}}+\sqrt{a_n}}$ =

38. The interior angles of a convex polygon are in A.P. The smallest angle is ${120}^{\circ }$ and the common difference is $5^{\circ }$. The number of its sides is

39. The H.M of the two numbers a and b is 4.The arithmetic mean A and geometric mean G satisfy the relation 2A + $G^2$ = 27. Then $a^2$ + $b^2$ =

40. If the $p^{\mathrm{th}}$, $q^{\mathrm{th}}$, $r^{\mathrm{th}}$ terms of a G.P are respectively a, b, c then $a^{q-r}$, $b^{r-p}$, $c^{p-q}$ =

41. The value of x+ y + z is 15 if a, x, y, z, b are in A.P, while the value of $\frac{1}{x}$+$\frac{1}{y}$+$\frac{1}{z}$ is $\frac{5}{8}$ if a, x, y, z, b are in H.P., then $a^2$+$b^2$ =

42. If a, b, c are in A.P and $a^2$, $b^2$, $c^2$ are in H.P, then $b^2$ =

43. $\frac{1^3}{1}$+ $\frac{1^3+2^3}{1+3}$$\frac{1^3+2^3+5^3}{1+3+5}$+......... to 16 terms=

44. Let a, b, c, d, e be five numbers sucht that a, b, c, are in A.P., b, c, d are in G.P. and c, d, e are in H.P. If a= 2 and e= 18, then b=

45. If $x_1$, $x_2$, $x_3$, ............., $x_n$ are in H.P, then $x_1$$x_2$ + $x_1{x}_3$ + $x_3{x}_4$ + ..........+ $x_{n-1}$$x_n$ =

46. The sum of the first n terms of the series 6+66+666+..........is

47. The ration of the sum of first 3 terms to the sum of first 6 terms of a G.P is 125 : 152. The common ration of the G.P is

48. There are four numbers of which the first three are in G.P and the last three are in A.P. If the first and the last number are equal, the sum of the four numbers is

49. The sum of 20 terms of the series 1 $\times$ $3^2$ + 2 $\times$ $5^2$ + 3 $\times$$7^2$ + ........... is

50. After sgtriking the floor, a ball rebounces ${\left(\frac{4}{5}\right)}^{\mathrm{th}}$ of its height from which it has fallen. If it is dropped from a height of 120m, the total distance it travels before coming to rest is