Progressions Test 2

Total Questions:50 Total Time: 75 Min

Remaining:

 

Questions 1 of 50

Question:If \({m^{th}}\) terms of the series \(63 + 65 + 67 + 69 + .........\) and \(3 + 10 + 17 + 24 + ......\) be equal, then \(m = \)

Answers Choices:

11

12

13

15

Questions 2 of 50

Question:The sum of 24 terms of the following series \(\sqrt 2 + \sqrt 8 + \sqrt {18} + \sqrt {32} + .........\) is

Answers Choices:

300

\(300\sqrt 2 \)

\(200\sqrt 2 \)

None of these

Questions 3 of 50

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

Answers Choices:

\(\frac{1}{{mn}}\)

\(\frac{1}{m} + \frac{1}{n}\)

1

0

Questions 4 of 50

Question:If \(1,\,\,{\log _9}({3^{1 - x}} + 2),\,\,{\log _3}({4.3^x} - 1)\) are in A.P. then x equals

Answers Choices:

\({\log _3}4\)

\(1 - {\log _3}4\)

\(1 - {\log _4}3\)

\({\log _4}3\)

Questions 5 of 50

Question:The sum of \(1 + 3 + 5 + 7 + .........\)upto \(n\) terms is

Answers Choices:

\({(n + 1)^2}\)

\({(2n)^2}\)

\({n^2}\)

\({(n - 1)^2}\)

Questions 6 of 50

Question:If the sum of the series \(54 + 51 + 48 + .............\) is 513, then the number of terms are

Answers Choices:

18

20

17

None of these

Questions 7 of 50

Question:The sum of the first and third term of an arithmetic progression is 12 and the product of first and second term is 24, then first term is

Answers Choices:

1

8

4

6

Questions 8 of 50

Question:If the sum of the first 2n terms of \(2,\,5,\,8...\) is equal to the sum of the first n terms of \(57,\,59,\,61...\), then n is equal to

Answers Choices:

10

12

11

13

Questions 9 of 50

Question:Three number are in A.P. such that their sum is 18 and sum of their squares is 158. The greatest number among them is

Answers Choices:

10

11

12

None of these

Questions 10 of 50

Question:If \(\frac{{3 + 5 + 7 + ..........{\rm{to}}\;n\;{\rm{terms}}}}{{5 + 8 + 11 + .........{\rm{to}}\;10\;{\rm{terms}}}} = 7\), then the value of\(n\) is

Answers Choices:

35

36

37

40

Questions 11 of 50

Question:If \({A_1},\,{A_2}\) be two arithmetic means between \(\frac{1}{3}\) and \(\frac{1}{{24}}\) , then their values are

Answers Choices:

\(\frac{7}{{72}},\,\frac{5}{{36}}\)

\(\frac{{17}}{{72}},\,\frac{5}{{36}}\)

\(\frac{7}{{36}},\,\frac{5}{{72}}\)

\(\frac{5}{{72}},\,\frac{{17}}{{72}}\)

Questions 12 of 50

Question:If the sum of three numbers of a arithmetic sequence is 15 and the sum of their squares is 83, then the numbers are

Answers Choices:

4, 5, 6

3, 5, 7

1, 5, 9

2, 5, 8

Questions 13 of 50

Question:The four arithmetic means between 3 and 23 are

Answers Choices:

5, 9, 11, 13

7, 11, 15, 19

5, 11, 15, 22

7, 15, 19, 21

Questions 14 of 50

Question:If the sum of three consecutive terms of an A.P. is 51 and the product of last and first term is 273, then the numbers are

Answers Choices:

21, 17, 13

20, 16, 12

22, 18, 14

24, 20, 16

Questions 15 of 50

Question:The terms of a G.P. are positive. If each term is equal to the sum of two terms that follow it, then the common ratio is

Answers Choices:

\(\frac{{1 - \sqrt 5 }}{2}\)

1

\(2\sqrt 5 \)

None of these

Questions 16 of 50

Question:If \(x,\,2x + 2,\,3x + 3,\) are in G.P., then the fourth term is

Answers Choices:

27

\( - 27\)

13.5

\( - 13.5\)

Questions 17 of 50

Question:If the ratio of the sum of first three terms and the sum of first six terms of a G.P. be 125 : 152, then the common ratio r is

Answers Choices:

\(\frac{3}{5}\)

\(\frac{5}{3}\)

\(\frac{2}{3}\)

\(\frac{3}{2}\)

Questions 18 of 50

Question:Fifth term of a G.P. is 2, then the product of its 9 terms is

Answers Choices:

256

512

1024

None of these

Questions 19 of 50

Question:If the sum of an infinite G.P. be 9 and the sum of first two terms be 5, then the common ratio is

Answers Choices:

1:3

3:2

3:4

2:3

Questions 20 of 50

Question:The sum of the first five terms of the series \(3 + 4\frac{1}{2} + 6\frac{3}{4} + ......\) will be

Answers Choices:

\(39\frac{9}{{16}}\)

\(18\frac{3}{{16}}\)

\(39\frac{7}{{16}}\)

\(13\frac{9}{{16}}\)

Questions 21 of 50

Question:The sum of few terms of any ratio series is 728, if common ratio is 3 and last term is 486, then first term of series will be

Answers Choices:

2

1

3

4

Questions 22 of 50

Question:The product of \(n\) positive numbers is unity. Their sum is

Answers Choices:

A positive integer

Equal to \(n + \frac{1}{n}\)

Divisible by \(n\)

Never less than

Questions 23 of 50

Question:Three numbers are in G.P. such that their sum is 38 and their product is 1728. The greatest number among them is

Answers Choices:

18

16

14

None of these

Questions 24 of 50

Question:If x, \({G_1}{,_\;}{G_2},\;y\)be the consecutive terms of a G.P., then the value of \({G_1}\,{G_2}\)will be

Answers Choices:

\(\frac{y}{x}\)

\(\frac{x}{y}\)

\(xy\)

\(\sqrt {xy} \)

Questions 25 of 50

Question:The sum of 3 numbers in geometric progression is 38 and their product is 1728. The middle number is

Answers Choices:

12

8

18

6

Questions 26 of 50

Question:If the product of three consecutive terms of G.P. is 216 and the sum of product of pair-wise is 156, then the numbers will be

Answers Choices:

1, 3, 9

2, 6, 18

3, 9, 27

2, 4, 8

Questions 27 of 50

Question:The sum of the series \(5.05 + 1.212 + 0.29088 + ...\,\infty \) is

Answers Choices:

6.93378

6.87342

6.74384

6.64474

Questions 28 of 50

Question:The sum of an infinite geometric series is 3. A series, which is formed by squares of its terms, have the sum also 3. First series will be

Answers Choices:

\(\frac{3}{2},\frac{3}{4},\frac{3}{8},\frac{3}{{16}},.....\)

\(\frac{1}{2},\frac{1}{4},\frac{1}{8},\frac{1}{{16}},.....\)

\(\frac{1}{3},\frac{1}{9},\frac{1}{{27}},\frac{1}{{81}},.....\)

\(1, - \frac{1}{3},\,\frac{1}{{{3^2}}}, - \frac{1}{{{3^3}}},.....\)

Questions 29 of 50

Question:If \({a^2} + a{b^2} + 16{c^2} = 2(3ab + 6bc + 4ac)\), where \(a,b,c\) are non-zero numbers. Then \(a,b,c\)are in

Answers Choices:

A.P

G.P

H.P

None of these

Questions 30 of 50

Question:The product (32)(32) 1/6(32)1/36 ...... to \(\infty \) is

Answers Choices:

16

32

64

0

62

Questions 31 of 50

Question:If \(H\) is the harmonic mean between \(p\) and \(q\), then the value of \(\frac{H}{p} + \frac{H}{q}\) is

Answers Choices:

2

\(\frac{{pq}}{{p + q}}\)

\(\frac{{p + q}}{{pq}}\)

None of these

Questions 32 of 50

Question:If the harmonic mean between \(a\) and \(b\) be \(H\), then the value of \(\frac{1}{{H - a}} + \frac{1}{{H - b}}\) is

Answers Choices:

\(a + b\)

\(ab\)

\(\frac{1}{a} + \frac{1}{b}\)

\(\frac{1}{a} - \frac{1}{b}\)

Questions 33 of 50

Question:If \({p^{th}},\;{q^{th}},\;{r^{th}}\) and \({s^{th}}\) terms of an A.P. be in G.P., then \((p - q),\;(q - r),\;(r - s)\) will be in

Answers Choices:

G.P.

A.P.

H.P.

None of these

Questions 34 of 50

Question:If the arithmetic and geometric means of a and b be \(A\) and \(G\) respectively, then the value of \(A - G\) will be

Answers Choices:

\(\frac{{a - b}}{a}\)

\(\frac{{a + b}}{2}\)

\({\left[ {\frac{{\sqrt a - \sqrt b }}{{\sqrt 2 }}} \right]^2}\)

\(\frac{{2ab}}{{a + b}}\)

Questions 35 of 50

Question:If \(a,\;b,\;c\)are in A.P., \(b,\;c,\;d\) are in G.P. and \(c,\;d,\;e\)are in H.P., then \(a,\;c,\;e\) are in

Answers Choices:

No particular order

A.P.

G.P.

H.P.

Questions 36 of 50

Question:If \(a,\;b,\;c\) are in G.P., \(a - b,\;c - a,\;b - c\)are in H.P., then \(a + 4b + c\)is equal to

Answers Choices:

0

\(1\)

\( - 1\)

None of these

Questions 37 of 50

Question:If \(\frac{{b + a}}{{b - a}} = \frac{{b + c}}{{b - c}}\), then\(a,\;b,\;c\) are in

Answers Choices:

A.P.

G.P.

H.P.

None of these

Questions 38 of 50

Question:If the ratio of two numbers be \(9:1\), then the ratio of geometric and harmonic means between them will be

Answers Choices:

\(1:9\)

\(5:3\)

\(3:5\)

\(2:5\)

Questions 39 of 50

Question:If \(a,\;b,\;c\) are in H.P., then for all \(n \in N\) the true statement is

Answers Choices:

\({a^n} + {c^n} < 2{b^n}\)

\({a^n} + {c^n} > 2{b^n}\)

\({a^n} + {c^n} = 2{b^n}\)

None of the above

Questions 40 of 50

Question:If A.M. of two terms is 9 and H.M. is 36, then G.M. will be

Answers Choices:

18

12

16

None of the above

Questions 41 of 50

Question:If \(p,\;q,\;r\) are in one geometric progression and \(a,\;b,\;c\) in another geometric progression, then \(cp,\;bq,\;ar\) are in

Answers Choices:

A.P.

H.P.

G.P.

None of these

Questions 42 of 50

Question:If first three terms of sequence \(\frac{1}{{16}},a,b,\frac{1}{6}\) are in geometric series and last three terms are in harmonic series, then the value of \(a\) and \(b\) will be

Answers Choices:

\(a = - \frac{1}{4},b = 1\)

\(a = \frac{1}{{12}},b = \frac{1}{9}\)

(1) and (2) both are true

None of these

Questions 43 of 50

Question:If \((y - x),\,\,2(y - a)\) and \((y - z)\) are in H.P., then \(x - a,\) \(y - a,\) \(z - a\) are in

Answers Choices:

A.P.

G.P.

H.P.

None of these

Questions 44 of 50

Question:If \(a,\,b,\,c\) are in A.P. and \({a^2},\,{b^2},{c^2}\)are in H.P., then

Answers Choices:

\(a \ne b \ne c\)

\({a^2} = {b^2} = \frac{{{c^2}}}{2}\)

\(a,\,b,\,c\) are in G.P.

\(\frac{{ - a}}{2},b,c\)are in G.P

Questions 45 of 50

Question:If \(a,\;b,\;c\) are in H.P., then which one of the following is true

Answers Choices:

\(\frac{1}{{b - a}} + \frac{1}{{b - c}} = \frac{1}{b}\)

\(\frac{{ac}}{{a + c}} = b\)

\(\frac{{b + a}}{{b - a}} + \frac{{b + c}}{{b - c}} = 1\)

None of these

Questions 46 of 50

Question:The sum of the series \(1 + \frac{{1.3}}{6} + \frac{{1.3.5}}{{6.8}} + ....\infty \)is

Answers Choices:

1

0

\(\infty \)

4

Questions 47 of 50

Question:The sum 1(1!) + 2(2!) + 3(3!) + ....+n (n!) equals

Answers Choices:

\(3\,(n\,!)\, + \,n - 3\)

\((n + 1)!\, - \,(n - 1)!\)

\((n + 1)\,!\, - 1\)

\(2\,(n\,!) - 2n - 1\)

Questions 48 of 50

Question:\(\frac{1}{{1.2}} + \frac{1}{{2.3}} + \frac{1}{{3.4}} + ........ + .......\frac{1}{{n.(n + 1)}}\)equals

Answers Choices:

\(\frac{1}{{n(n + 1)}}\)

\(\frac{n}{{n + 1}}\)

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

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

Questions 49 of 50

Question:Sum of the series \(\frac{2}{3} + \frac{8}{9} + \frac{{26}}{{27}} + \frac{{80}}{{81}} + .....\) to n terms is

Answers Choices:

\(n - \frac{1}{2}({3^n} - 1)\)

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

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

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

Questions 50 of 50

Question:\(\sum\limits_{m = 1}^n {{m^2}} \) is equal to

Answers Choices:

\(\frac{{m(m + 1)}}{2}\)

\(\frac{{m(m + 1)(2m + 1)}}{6}\)

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

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