# Progressions Test 1

Total Questions:50 Total Time: 75 Min

Remaining:

## Questions 1 of 50

Question:If $$\tan \,n\theta = \tan m\theta$$, then the different values of $$\theta$$  will be in

A.P.

G.P.

H.P.

None of these

## Questions 2 of 50

Question:$${n^{th}}$$ term of the series $$3.8 + 6.11 +$$ $$9.14 + 12.17 + .....$$will be

$$3n(3n + 5)$$

$$3n(n + 5)$$

$$n(3n + 5)$$

$$n(n + 5)$$

## Questions 3 of 50

Question:The sum of integers from 1 to 100 that are divisible by 2 or 5 is

3000

3050

4050

None of these

## Questions 4 of 50

Question:If $$p$$times the $${p^{th}}$$ term of an A.P. is equal to $$q$$ times the $${q^{th}}$$ term of an A.P., then $${(p + q)^{th}}$$ term is

0

1

2

3

## Questions 5 of 50

Question:The sums of $$n$$ terms of two arithmatic series are in the ratio $$2n + 3:6n + 5$$, then the ratio of their $${13^{th}}$$ terms is

53 : 155

27 : 77

29 : 83

31 : 89

## Questions 6 of 50

Question:If $${a_{\rm{m}}}$$ denotes the $${m^{{\rm{th}}}}$$ term of an A.P. then $${a_m}$$ =

$$\frac{2}{{{a_{m + k}} + {a_{m - k}}}}$$

$$\frac{{{a_{m + k}} - {a_{m - k}}}}{2}$$

$$\frac{{{a_{m + k}} + {a_{m - k}}}}{2}$$

None of these

## Questions 7 of 50

Question:If $${a_1},\;{a_2},............,{a_n}$$ are in A.P. with common difference , $$d$$, then the sum of the following series is $$\sin d\left( {\cos ec{a_1}.\cos ec{a_2} + \cos ec{a_2}.\cos ec{a_3}} \right) + ...... + \cos ec\,{a_{n - 1}}\cos ec\,{a_n}$$

$$\sec {a_1} - \sec {a_n}$$

$$\cot {a_1} - \cot {a_n}$$

$$\tan {a_1} - \tan {a_n}$$

$$c{\rm{osec}}\;{a_1} - {\rm{cosec}}\;{a_n}$$

## Questions 8 of 50

Question:If the sum of the series $$2 + 5 + 8 + 11............$$is 60100, then the number of terms are

100

200

150

250

## Questions 9 of 50

Question:The sum of all natural numbers between 1 and 100 which are multiples of 3 is

1680

1683

1681

1682

## Questions 10 of 50

Question:If $${S_k}$$denotes the sum of first $$k$$terms of an arithmetic progression whose first term and common difference are $$a$$and $$d$$ respectively, then $${S_{kn}}/{S_n}$$be independent of $$n$$if

$$2a - d = 0$$

$$a - d = 0$$

$$a - 2d = 0$$

None of these

## Questions 11 of 50

Question:A series whose nth term is $$\left( {\frac{n}{x}} \right) + y,$$the sum of r terms will be

$$\left\{ {\frac{{r(r + 1)}}{{2x}}} \right\} + ry$$

$$\left\{ {\frac{{r(r - 1)}}{{2x}}} \right\}$$

$$\left\{ {\frac{{r(r - 1)}}{{2x}}} \right\} - ry$$

$$\left\{ {\frac{{r(r + 1)}}{{2y}}} \right\} - rx$$

## Questions 12 of 50

Question:The sum of the integers from 1 to 100 which are not divisible by 3 or 5 is

2489

4735

2317

2632

## Questions 13 of 50

Question:There are 15 terms in an arithmetic progression. Its first term is 5 and their sum is 390. The middle term is

23

26

29

32

## Questions 14 of 50

Question:If the sum of the 10 terms of an A.P. is 4 times to the sum of its 5 terms, then the ratio of first term and common difference is

$$1:2$$

$$2:1$$

$$2:3$$

$$3:2$$

## Questions 15 of 50

Question:Three numbers are in A.P. whose sum is 33 and product is 792, then the smallest number from these numbers is

4

8

11

14

## Questions 16 of 50

Question:If $$a,\;b,\;c,\;d,\;e,\;f$$ are in A.P., then the value of $$e - c$$ will be

$$2(c - a)$$

$$2(f - d)$$

$$2(d - c)$$

$$d - c$$

## Questions 17 of 50

Question:The number which should be added to the numbers 2, 14, 62 so that the resulting numbers may be in G.P., is

1

2

3

4

## Questions 18 of 50

Question:If $${(p + q)^{th}}$$ term of a G.P. be $$m$$ and $${(p - q)^{th}}$$term be $$n$$, then the $${p^{th}}$$ term will be [RPET 1997; MP PET 1985, 99]

$$m/n$$

$$\sqrt {mn}$$

$$mn$$

0

## Questions 19 of 50

Question:If the nth term of geometric progression $$5, - \frac{5}{2},\frac{5}{4}, - \frac{5}{8},...$$ is $$\frac{5}{{1024}}$$, then the value of n is

11

10

9

4

## Questions 20 of 50

Question:The third term of a G.P. is the square of first term. If the second term is 8, then the $${6^{th}}$$ term is

120

124

128

132

## Questions 21 of 50

Question:The solution of the equation $$1 + a + {a^2} + {a^3} + ....... + {a^x}$$ $$= (1 + a)(1 + {a^2})(1 + {a^4})$$ is given by $$x$$ is equal to

3

5

7

None of these

## Questions 22 of 50

Question:If in a geometric progression $$\left\{ {{a_n}} \right\},\;{a_1} = 3,\;{a_n} = 96$$ and $${S_n} = 189$$ then the value of $$n$$ is

5

6

7

8

## Questions 23 of 50

Question:The two geometric means between the number 1 and 64 are

1 and 64

4 and 16

2 and 16

8 and 16

## Questions 24 of 50

Question:If $$a,\;b,\;c$$ are in G.P., then

$${a^2},\;{b^2},\;{c^2}$$are in G.P.

$${a^2}(b + c),\;{c^2}(a + b),\;{b^2}(a + c)$$ are in G.P.

$$\frac{a}{{b + c}},\;\frac{b}{{c + a}},\;\frac{c}{{a + b}}$$ are in G.P.

None of the above

## Questions 25 of 50

Question:If $$S$$ is the sum to infinity of a G.P., whose first term is $$a$$, then the sum of the first $$n$$ terms is

$$S{\left( {1 - \frac{a}{S}} \right)^n}$$

$$S\left[ {1 - {{\left( {1 - \frac{a}{S}} \right)}^n}} \right]$$

$$a\left[ {1 - {{\left( {1 - \frac{a}{S}} \right)}^n}} \right]$$

None of these

## Questions 26 of 50

Question:0.14189189189,....................... can be expressed as a rational number

$$\frac{7}{{3700}}$$

$$\frac{7}{{50}}$$

$$\frac{{525}}{{111}}$$

$$\frac{{21}}{{148}}$$

## Questions 27 of 50

Question:If s is the sum of an infinite G.P., the first term a then the common ratio r given by

$$\frac{{a - s}}{s}$$

$$\frac{{s - a}}{s}$$

$$\frac{a}{{1 - s}}$$

$$\frac{{s - a}}{a}$$

## Questions 28 of 50

Question:The sum to infinity of the progression $$9 - 3 + 1 - \frac{1}{3} + .....$$ is

9

2-Sep

27/4

15/2

## Questions 29 of 50

Question:n a H.P., pth term is q and the qth term is p. Then pqth term is

0

1

pq

$$pq(p + q)$$

## Questions 30 of 50

Question:The 4th term of a H.P. is $$\frac{3}{5}$$ and 8th term is $$\frac{1}{3},$$ then its 6th term is

$$\frac{1}{6}$$

$$\frac{3}{7}$$

$$\frac{1}{7}$$

$$\frac{3}{5}$$

## Questions 31 of 50

Question:If $${\log _a}x,\;{\log _b}x,\;{\log _c}x$$ be in H.P., then $$a,\;b,\;c$$ are in

A.P.

H.P.

G.P.

None of these

## Questions 32 of 50

Question:If three numbers be in G.P., then their logarithms will be in

A.P.

G.P.

H.P.

None of these

## Questions 33 of 50

Question:If $$a,\;b,\;c$$ are in A.P. and $$|a|,\;|b|,\;|c|\; < 1$$ and $$x = 1 + a + {a^2} + ........\infty$$$$y = 1 + b + {b^2} + .......\infty$$$$z = 1 + c + {c^2}........\infty$$. Then$$x,\;y,\;z$$ shall be in

A.P.

G.P.

H.P.

None of these

## Questions 34 of 50

Question:If three unequal non-zero real numbers $$a,\;b,\;c$$are in G.P. and $$b - c,\;c - a,\;a - b$$are in H.P., then the value of $$a + b + c$$ is independent of

$$a$$

$$b$$

$$c$$

None of these

## Questions 35 of 50

Question:The numbers $$(\sqrt 2 + 1),\;1,\;(\sqrt 2 - 1)$$ will be in

A.P.

G.P.

H.P.

None of these

## Questions 36 of 50

Question:If the ratio of H.M. and G.M. of two quantities is $$12:13$$, then the ratio of the numbers is

$$1:2$$

$$2:3$$

$$3:4$$

None of these

## Questions 37 of 50

Question:If $$\frac{{a + bx}}{{a - bx}} = \frac{{b + cx}}{{b - cx}} = \frac{{c + dx}}{{c - dx}}(x \ne 0)$$, then $$a,\;b,\;c,\;d$$ are in

A.P.

G.P.

H.P.

None of these

## Questions 38 of 50

Question:If $$a,\;b,\;c$$ are in A.P. and $$a,\;c - b,\;b - a$$ are in G.P. $$(a \ne b \ne c)$$, then $$a:b:c$$ is

$$1:3:5$$

$$1:2:4$$

$$1:2:3$$

None of these

## Questions 39 of 50

Question:If a,b,c are in A.P., then $$\frac{1}{{\sqrt a + \sqrt b }},\,\frac{1}{{\sqrt a + \sqrt c }},$$ $$\frac{1}{{\sqrt b + \sqrt c }}$$ are in

A.P.

G.P.

H.P.

None of these

## Questions 40 of 50

Question:The sum of three decreasing numbers in A.P. is 27. If $$- 1,\, - 1,\,3$$ are added to them respectively, the resulting series is in G.P. The numbers are

5, 9, 13

15, 9, 3

13, 9, 5

17, 9, 1

## Questions 41 of 50

Question:If $$a,b,c$$are in G.P. then $${\log _a}x,{\log _b}x,{\log _c}x$$ are in

A.P.

G.P.

H.P.

None of these

## Questions 42 of 50

Question:If $$a,\,b,\,c$$ are three unequal numbers such that $$a,\,b,\,c$$ are in A.P. and b - a, c - b, a are in G.P., then a : b : c is

1:2:3

2:3:1

1:3:2

3:2:1

## Questions 43 of 50

Question:$$2 + 4 + 7 + 11 + 16 + ......$$to $$n$$ terms =

$$\frac{1}{6}({n^2} + 3n + 8)$$

$$\frac{n}{6}({n^2} + 3n + 8)$$

$$\frac{1}{6}({n^2} - 3n + 8)$$

$$\frac{n}{6}({n^2} - 3n + 8)$$

## Questions 44 of 50

Question:Sum of n terms of series $$12 + 16 + 24 + 40 + .....$$ will be

$$2\,({2^n} - 1) + 8n$$

$$2({2^n} - 1) + 6n$$

$$3({2^n} - 1) + 8n$$

$$4({2^n} - 1) + 8n$$

## Questions 45 of 50

Question:The sum of the series $$1 + (1 + 2) + (1 + 2 + 3) + ............$$upto $$n$$ terms, will be

$${n^2} - 2n + 6$$

$$\frac{{n(n + 1)(2n - 1)}}{6}$$

$${n^2} + 2n + 6$$

$$\frac{{n(n + 1)(n + 2)}}{6}$$

## Questions 46 of 50

Question:The sum of $$n$$ terms of the series whose $${n^{hth}}$$ term is $$n(n + 1)$$ is equal to

$$\frac{{n(n + 1)(n + 2)}}{3}$$

$$\frac{{(n + 1)(n + 2)(n + 3)}}{{12}}$$

$${n^2}(n + 2)$$

$$n(n + 1)(n + 2)$$

## Questions 47 of 50

Question:Sum of the $$n$$ terms of the series $$\frac{3}{{{1^2}}} + \frac{5}{{{1^2} + {2^2}}} + \frac{7}{{{1^2} + {2^2} + {3^2}}}\, + ...\,\,{\rm{is}}$$

$$\frac{{2n}}{{n + 1}}$$

$$\frac{{4n}}{{n + 1}}$$

$$\frac{{6n}}{{n + 1}}$$

$$\frac{{9n}}{{n + 1}}$$

## Questions 48 of 50

Question:The sum of the series $$1.3.5 + .2.5.8 + 3.7.11 + .........$$upto $$'n'$$ terms is

$$\frac{{n\,(n + 1)(9{n^2} + 23n + 13)}}{6}$$

$$\frac{{n\,(n - 1)(9{n^2} + 23n + 12)}}{6}$$

$$\frac{{(n + 1)(9{n^2} + 23n + 13)}}{6}$$

$$\frac{{n\,(9{n^2} + 23n + 13)}}{6}$$

## Questions 49 of 50

Question:If $$\frac{1}{{{1^4}}} + \frac{1}{{{2^4}}} + \frac{1}{{{3^4}}} + ..... + \infty = \frac{{{\pi ^4}}}{{90}}$$, then the value of $$\frac{1}{{{1^4}}} + \frac{1}{{{3^4}}} + \frac{1}{{{5^4}}} + .....\infty$$is

$$\frac{{{\pi ^4}}}{{96}}$$

$$\frac{{{\pi ^4}}}{{45}}$$

$$\frac{{89}}{{90}}{\pi ^4}$$

None of these

## Questions 50 of 50

Question:The value of $$\frac{1}{{(1 + a)(2 + a)}} + \frac{1}{{(2 + a)(3 + a)}}$$ $$\frac{1}{{(3 + a)(4 + a)}}$$+ ..... +$$\infty$$ is, (where a is a constant)

$$\frac{1}{{1 + a}}$$
$$\frac{2}{{1 + a}}$$
$$\infty$$