Progressions Test 7

Total Questions:50 Total Time: 75 Min

Remaining:

 

Questions 1 of 50

Question:The number of terms in the series \(101 + 99 + 97 + ..... + 47\) is

Answers Choices:

25

28

30

20

Questions 2 of 50

Question:If the \({p^{th}}\) term of an A.P. be \(q\) and \({q^{th}}\)term be p, then its \({r^{th}}\) term will be

Answers Choices:

\(p + q + r\)

\(p + q - r\)

\(p + r - q\)

\(p - q - r\)

Questions 3 of 50

Question:If the numbers \(a,\;b,\;c,\;d,\;e\)form an A.P., then the value of \(a - 4b + 6c - 4d + e\) is

Answers Choices:

1

2

0

None of these

Questions 4 of 50

Question:The sixth term of an A.P. is equal to 2, the value of the common difference of the A.P. which makes the product \({a_1}{a_4}{a_5}\) least is given by

Answers Choices:

\(x = \frac{8}{5}\)

\(x = \frac{5}{4}\)

\(x = 2/3\)

None of these

Questions 5 of 50

Question:The ratio of the sums of first \(n\)even numbers and \(n\) odd numbers will be

Answers Choices:

\(1:n\)

\((n + 1):1\)

\((n + 1):n\)

\((n - 1):1\)

Questions 6 of 50

Question:If \({a_1},\;{a_2},\;{a_3}.......{a_n}\) are in A.P., where \({a_i} > 0\) for all \(i\), then the value of \(\frac{1}{{\sqrt {{a_1}} + \sqrt {{a_2}} }} + \frac{1}{{\sqrt {{a_2}} + \sqrt {{a_3}} }} + \) \(........ + \frac{1}{{\sqrt {{a_{n - 1}}} + \sqrt {{a_n}} }} = \)

Answers Choices:

\(\frac{{n - 1}}{{\sqrt {{a_1}} + \sqrt {{a_n}} }}\)

\(\frac{{n + 1}}{{\sqrt {{a_1}} + \sqrt {{a_n}} }}\)

\(\frac{{n - 1}}{{\sqrt {{a_1}} - \sqrt {{a_n}} }}\)

\(\frac{{n + 1}}{{\sqrt {{a_1}} - \sqrt {{a_n}} }}\)

Questions 7 of 50

Question:If \({S_n}\) denotes the sum of \(n\) terms of an arithmetic progression, then the value of \(({S_{2n}} - {S_n})\)is equal to

Answers Choices:

\(2{S_n}\)

\({S_{3n}}\)

\(\frac{1}{3}{S_{3n}}\)

\(\frac{1}{2}{S_n}\)

Questions 8 of 50

Question:The solution of \({\log _{\sqrt 3 }}x + {\log _{\sqrt[4]{3}}}x + {\log _{\sqrt[6]{3}}}x + ......... + {\log _{\sqrt[{16}]{3}}}x = 36\) is

Answers Choices:

\(x = 3\)

\(x = 4\sqrt 3 \)

\(x = 9\)

\(x = \sqrt 3 \)

Questions 9 of 50

Question:The sum of the first four terms of an A.P. is 56. The sum of the last four terms is 112. If its first term is 11, the number of terms is

Answers Choices:

10

11

12

None of these

Questions 10 of 50

Question:The number of terms of the A.P. 3,7,11,15...to be taken so that the sum is 406 is

Answers Choices:

5

10

12

14

Questions 11 of 50

Question:If the sum of two extreme numbers of an A.P. with four terms is 8 and product of remaining two middle term is 15, then greatest number of the series will be

Answers Choices:

5

7

9

11

Questions 12 of 50

Question:If the sides of a right angled traingle are in A.P., then the sides are proportional to

Answers Choices:

1:02:03

2:03:04

3:04:05

4:05:06

Questions 13 of 50

Question:If the \({4^{th}},\;{7^{th}}\) and \({10^{th}}\) terms of a G.P. be \(a,\;b,\;c\) respectively, then the relation between [a,\;b,\;c\)is

Answers Choices:

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

\({a^2} = bc\)

\({b^2} = ac\)

\({c^2} = ab\)

Questions 14 of 50

Question:If the first term of a G.P. be 5 and common ratio be \( - 5\), then which term is 3125

Answers Choices:

\({6^{th}}\)

\({5^{th}}\)

\({7^{th}}\)

\({8^{th}}\)

Questions 15 of 50

Question:If the \({10^{th}}\) term of a geometric progression is 9 and \({4^{th}}\) term is 4, then its \({7^{th}}\) term is

Answers Choices:

6

36

\(\frac{4}{9}\)

\(\frac{9}{4}\)

Questions 16 of 50

Question:The 6th term of a G.P. is 32 and its 8th term is 128, then the common ratio of the G.P. is

Answers Choices:

1

2

4

4

Questions 17 of 50

Question:The number \(111..............1\) (91 times) is a

Answers Choices:

Even number

Prime number

Not prime

None of these

Questions 18 of 50

Question:For a sequence \( < {a_n} > ,\;{a_1} = 2\) and \(\frac{{{a_{n + 1}}}}{{{a_n}}} = \frac{1}{3}\). Then \(\sum\limits_{r = 1}^{20} {{a_r}} \) is

Answers Choices:

\(\frac{{20}}{2}[4 + 19 \times 3]\)

\(3\left( {1 - \frac{1}{{{3^{20}}}}} \right)\)

\(2(1 - {3^{20}})\)

None of these

Questions 19 of 50

Question:The G.M. of the numbers \(3,\,{3^2},\,{3^3},....,\,{3^n}\) is

Answers Choices:

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

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

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

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

Questions 20 of 50

Question:The product of three geometric means between 4 and \(\frac{1}{4}\) will be

Answers Choices:

4

2

\( - 1\)

1

Questions 21 of 50

Question:The sum of infinite terms of a G.P. is \(x\) and on squaring the each term of it, the sum will be \(y\), then the common ratio of this series is

Answers Choices:

\(\frac{{{x^2} - {y^2}}}{{{x^2} + {y^2}}}\)

\(\frac{{{x^2} + {y^2}}}{{{x^2} - {y^2}}}\)

\(\frac{{{x^2} - y}}{{{x^2} + y}}\)

\(\frac{{{x^2} + y}}{{{x^2} - y}}\)

Questions 22 of 50

Question:If the sum of an infinite G.P. and the sum of square of its terms is 3, then the common ratio of the first series is

Answers Choices:

1

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

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

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

Questions 23 of 50

Question:The value of \(\overline {0.037} \) where, \(\overline {.037} \) stands for the number 0.037037037........ is

Answers Choices:

\(\frac{{37}}{{1000}}\)

\(\frac{1}{{27}}\)

\(\frac{1}{{37}}\)

\(\frac{{37}}{{999}}\)

Questions 24 of 50

Question:If \(x\) is added to each of numbers 3, 9, 21 so that the resulting numbers may be in G.P., then the value of \(x\) will be

Answers Choices:

3

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

2

\(\frac{1}{3}\)

Questions 25 of 50

Question:If the \({7^{th}}\) term of a harmonic progression is 8 and the \({8^{th}}\)term is 7, then its \({15^{th}}\) term is

Answers Choices:

16

14

\(\frac{{27}}{{14}}\)

\(\frac{{56}}{{15}}\)

Questions 26 of 50

Question:If the \({7^{th}}\) term of a H.P. is \(\frac{1}{{10}}\) and the \({12^{th}}\) term is \(\frac{1}{{25}}\), then the \({20^{th}}\) term is

Answers Choices:

\(\frac{1}{{37}}\)

\(\frac{1}{{41}}\)

\(\frac{1}{{45}}\)

\(\frac{1}{{49}}\)

Questions 27 of 50

Question:If sixth term of a H.P. is \(\frac{1}{{61}}\) and its tenth term is \(\frac{1}{{105}},\) then first term of that H.P. is

Answers Choices:

\(\frac{1}{{28}}\)

\(\frac{1}{{39}}\)

\(\frac{1}{6}\)

\(\frac{1}{{17}}\)

Questions 28 of 50

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

Answers Choices:

\(ab = cd\)

\(ad = bc\)

\(ac = bd\)

\(abcd = 1\)

Questions 29 of 50

Question:If \(a,\;b,\;c\) are in A.P., then\(\frac{a}{{bc}},\;\frac{1}{c},\;\frac{2}{b}\) are in

Answers Choices:

A.P.

G.P.

H.P.

None of these

Questions 30 of 50

Question:If the roots of\(a\,(b - c){x^2} + b\,(c - a)x + c\,(a - b) = 0\) be equal, then \(a,\;b,\;c\)are in

Answers Choices:

A.P.

G.P.

H.P.

None of these

Questions 31 of 50

Question:If \({a^2},\;{b^2},\;{c^2}\) are in A.P., then \({(b + c)^{ - 1}},\;{(c + a)^{ - 1}}\) and \({(a + b)^{ - 1}}\) will be in

Answers Choices:

H.P.

G.P.

A.P.

None of these

Questions 32 of 50

Question:If \(a,\;b,\;c\) are in A.P., then \(\frac{1}{{bc}},\;\frac{1}{{ca}},\;\frac{1}{{ab}}\) will be in

Answers Choices:

A.P.

G.P.

H.P.

None of these

Questions 33 of 50

Question:If \(x,\;1,\;z\) are in A.P. and \(x,\;2,\;z\) are in G.P., then \(x,\;4,\;z\) will be in

Answers Choices:

A.P.

G.P.

H.P.

None of these

Questions 34 of 50

Question:If the \({p^{th}},\;{q^{th}}\) and \({r^{th}}\)term of a G.P. and H.P. are \(a,\;b,\;c\), then \(a(b - c)\log a + b(c - a)\) \(\log b + c(a - b)\log c = \)

Answers Choices:

\( - 1\)

0

1

Does not exist

Questions 35 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:

\(a = b = c\)

\(2b = 3a + c\)

\({b^2} = \sqrt {(ac/8)} \)

None of these

Questions 36 of 50

Question:In the four numbers first three are in G.P. and last three are in A.P. whose common difference is 6. If the first and last numbers are same, then first will be

Answers Choices:

2

4

6

8

Questions 37 of 50

Question:If \({\log _x}y,\;{\log _z}x,\;{\log _y}z\) are in G.P. \(xyz = 64\) and \({x^3},\;{y^3},\;{z^3}\) are in A.P., then

Answers Choices:

\(x = y = z\)

\(x = 4\)

\(x,\;y,\,z\)are in G.P.

All the above

Questions 38 of 50

Question:If three unequal numbers \(p,\;q,\;r\) are in H.P. and their squares are in A.P., then the ratio \(p:q:r\) is

Answers Choices:

\(1 - \sqrt 3 :2:1 + \sqrt 3 \)

\(1:\sqrt 2 : - \sqrt 3 \)

\(1: - \sqrt 2 :\sqrt 3 \)

\(1 \mp \sqrt 3 : - 2:1 \pm \sqrt 3 \)

Questions 39 of 50

Question:If \({G_1}\) and \({G_2}\) are two geometric means and A the arithmetic mean inserted between two numbers, then the value of \(\frac{{G_1^2}}{{{G_2}}} + \frac{{G_2^2}}{{{G_1}}}\) is

Answers Choices:

\(\frac{A}{2}\)

A

2:00 AM

None of these

Questions 40 of 50

Question:If \(\log (x + z) + \log (x + z - 2y) = 2\log (x - z),\,\) then \(x,\,y,\,z\) are in

Answers Choices:

H.P.

G.P.

A.P.

None of these

Questions 41 of 50

Question:If A and G are arithmetic and geometric means and \({x^2} - 2Ax + {G^2} = 0\), then

Answers Choices:

\(A = G\)

\(A > G\)

\(A < G\)

\(A = - \,G\)

Questions 42 of 50

Question:If ln \((a + c)\), In \((c - a)\), In \((a - 2b + c)\) are in A.P., then

Answers Choices:

\(a,\;b,\;c\)are in A.P.

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

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

\(a,\;b,\;c\) are in H.P

Questions 43 of 50

Question:The sum of infinite terms of the following series \(1 + \frac{4}{5} + \frac{7}{{{5^2}}} + \frac{{10}}{{{5^3}}} + .........\) will be

Answers Choices:

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

\(\frac{{35}}{8}\)

\(\frac{{35}}{4}\)

\(\frac{{35}}{{16}}\)

Questions 44 of 50

Question:The sum of the series \(1 + 3x + 6{x^2} + 10{x^3} + ........\infty \) will be

Answers Choices:

\(\frac{1}{{{{(1 - x)}^2}}}\)

\(\frac{1}{{1 - x}}\)

\(\frac{1}{{{{(1 + x)}^2}}}\)

\(\frac{1}{{{{(1 - x)}^3}}}\)

Questions 45 of 50

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

Answers Choices:

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

\(\frac{{{n^2}(n + 1)}}{4}\)

\(\frac{{n\,(n - 1)\,(2n - 1)}}{6}\)

\({n^2}\)

Questions 46 of 50

Question:The sum of first \(n\) terms of the given series \({1^2} + {2.2^2} + {3^2} + {2.4^2} + {5^2} + {2.6^2} + ............\)is\(\frac{{n{{(n + 1)}^2}}}{2}\), when \(n\) is even. When \(n\) is odd, the sum will be

Answers Choices:

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

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

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

None of these

Questions 47 of 50

Question:The sum of all the products of the first \(n\) natural numbers taken two at a time is

Answers Choices:

\(\frac{1}{{24}}n(n - 1)(n + 1)(3n + 2)\)

\(\frac{{{n^2}}}{{48}}(n - 1)(n - 2)\)

\(\frac{1}{6}n(n + 1)(n + 2)(n + 5)\)

None of these

Questions 48 of 50

Question:The sum of the series \({1.3^2} + {2.5^2} + {3.7^2} + ..........\)upto \(20\) terms is

Answers Choices:

188090

189080

199080

None of these

Questions 49 of 50

Question:If the sum of \(1 + \frac{{1 + 2}}{2} + \frac{{1 + 2 + 3}}{3} + .....\) to n terms is S, then S is equal to

Answers Choices:

\(\frac{{n(n + 3)}}{4}\)

\(\frac{{n(n + 2)}}{4}\)

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

\({n^2}\)

Questions 50 of 50

Question:The nth term of the series \(\frac{2}{{1!}} + \frac{7}{{2\,!}} + \frac{{15}}{{3\,!}} + \frac{{26}}{{4\,!}} + .....\) is

Answers Choices:

\(\frac{{n\,(3n - 1)}}{{2(n)\,!}}\)

\(\frac{{n\,(3n + 1)}}{{2\,(n)\,!}}\)

\(\frac{n}{2}\frac{{3n}}{{(n)\,!}}\)

None of these