Progressions Test 4

Total Questions:50 Total Time: 75 Min

Remaining:

 

Questions 1 of 50

Question:If the \({9^{th}}\)term of an A.P. is 35 and \({19^{th}}\) is 75, then its \({20^{th}}\) terms will be

Answers Choices:

78

79

80

81

Questions 2 of 50

Question:The \({9^{th}}\) term of the series \(27 + 9 + 5\frac{2}{5} + 3\frac{6}{7} + ........\) will be

Answers Choices:

\(1\frac{{10}}{{17}}\)

\(\frac{{10}}{{17}}\)

\(\frac{{16}}{{27}}\)

\(\frac{{17}}{{27}}\)

Questions 3 of 50

Question:The sum of the series \(\frac{1}{2} + \frac{1}{3} + \frac{1}{6} + ........\)to 9 terms is

Answers Choices:

\( - \frac{5}{6}\)

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

1

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

Questions 4 of 50

Question:The interior angles of a polygon are in A.P. If the smallest angle be \({120^o}\)and the common difference be 5o, then the number of sides is

Answers Choices:

8

10

9

6

Questions 5 of 50

Question:If the first term of an A.P. be 10, last term is 50 and the sum of all the terms is 300, then the number of terms are

Answers Choices:

5

8

10

15

Questions 6 of 50

Question:The maximum sum of the series \(20 + 19\frac{1}{3} + 18\frac{2}{3} + .........\) is

Answers Choices:

310

300

320

None of these

Questions 7 of 50

Question:If \({a_1},\,{a_2},....,{a_{n + 1}}\)are in A.P., then \(\frac{1}{{{a_1}{a_2}}} + \frac{1}{{{a_2}{a_3}}} + ..... + \frac{1}{{{a_n}{a_{n + 1}}}}\) is

Answers Choices:

\(\frac{{n - 1}}{{{a_1}{a_{n + 1}}}}\)

\(\frac{1}{{{a_1}{a_{n + 1}}}}\)

\(\frac{{n + 1}}{{{a_1}{a_{n + 1}}}}\)

\(\frac{n}{{{a_1}{a_{n + 1}}}}\)

Questions 8 of 50

Question:If the sum of the first \(n\)terms of a series be \(5{n^2} + 2n\), then its second term is

Answers Choices:

7

17

24

42

Questions 9 of 50

Question:If \(A\) be an arithmetic mean between two numbers and \(S\) be the sum of \(n\) arithmetic means between the same numbers, then

Answers Choices:

\(S = n\,A\)

\(A = n\,S\)

\(A = S\)

None of these

Questions 10 of 50

Question:The arithmetic mean of first n natural number

Answers Choices:

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

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

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

\(n\)

Questions 11 of 50

Question:The difference between an integer and its cube is divisible by

Answers Choices:

4

6

9

None of these

Questions 12 of 50

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

Answers Choices:

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

abc

2 abc

4 abc

Questions 13 of 50

Question:If the third term of a G.P. is 4 then the product of its first 5 terms is

Answers Choices:

\({4^3}\)

\({4^4}\)

\({4^5}\)

None of these

Questions 14 of 50

Question:If the \({5^{th}}\) term of a G.P. is \(\frac{1}{3}\) and \({9^{th}}\) term is \(\frac{{16}}{{243}}\), then the \({4^{th}}\) term will be

Answers Choices:

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

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

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

\(\frac{2}{5}\)

Questions 15 of 50

Question:If the sum of three terms of G.P. is 19 and product is 216, then the common ratio of the series is [Roorkee 1972]

Answers Choices:

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

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

2

3

Questions 16 of 50

Question:The sum of the series \(6 + 66 + 666 + ..........\)upto \(n\) terms is

Answers Choices:

\(({10^{n - 1}} - 9n + 10)/81\)

\(2({10^{n + 1}} - 9n - 10)/27\)

\(2({10^n} - 9n - 10)/27\)

None of these

Questions 17 of 50

Question:The sum of a G.P. with common ratio 3 is 364, and last term is 243, then the number of terms is

Answers Choices:

6

5

4

10

Questions 18 of 50

Question:If \(n\) geometric means be inserted between \(a\) and \(b\)then the \({n^{th}}\) geometric mean will be

Answers Choices:

\(a\,{\left( {\frac{b}{a}} \right)^{\frac{n}{{n - 1}}}}\)

\(a\,{\left( {\frac{b}{a}} \right)^{\frac{{n - 1}}{n}}}\)

\(a\,{\left( {\frac{b}{a}} \right)^{\frac{n}{{n + 1}}}}\)

\(a\,{\left( {\frac{b}{a}} \right)^{\frac{1}{n}}}\)

Questions 19 of 50

Question:The sum can be found of a infinite G.P. whose common ratio is \(r\)

Answers Choices:

For all values of \(r\)

For only positive value of \(r\)

Only for \(0 < r < 1\)

Only for \( - 1 < r < 1(r \ne 0)\)

Questions 20 of 50

Question:If \(A = 1 + {r^z} + {r^{2z}} + {r^{3z}} + .......\infty \), then the value of r will be

Answers Choices:

\(A{(1 - A)^z}\)

\({\left( {\frac{{A - 1}}{A}} \right)^{1/z}}\)

\({\left( {\frac{1}{A} - 1} \right)^{1/z}}\)

\(A{(1 - A)^{1/z}}\)

Questions 21 of 50

Question:If \(y = x + {x^2} + {x^3} + .......\,\infty ,\,{\rm{then}}\,\,x = \)

Answers Choices:

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

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

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

None of these

Questions 22 of 50

Question:If sum of infinite terms of a G.P. is 3 and sum of squares of its terms is 3, then its first term and common ratio are

Answers Choices:

3/2, 1/2

1, ½

3/2, 2

None of these

Questions 23 of 50

Question:If \({a_1},\;{a_2},\;{a_3},...............,\;{a_n}\) are in H.P., then \({a_1}{a_2} + {a_2}{a_3} + \) \(.......... + {a_{n - 1}}{a_n}\) will be equal to

Answers Choices:

\({a_1}{a_n}\)

\(n{a_1}{a_n}\)

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

None of these

Questions 24 of 50

Question:If \(x,\;y,\;z\) are in H.P., then the value of expression \(\log (x + z) + \log (x - 2y + z)\) will be

Answers Choices:

\(\log (x - z)\)

\(2\log (x - z)\)

\(3\log (x - z)\)

\(4\log (x - z)\)

Questions 25 of 50

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

Answers Choices:

1

\( - 1\)

0

2

Questions 26 of 50

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

Answers Choices:

4

2

1

\(a + b\)

Questions 27 of 50

Question:If \(\frac{1}{{b - a}} + \frac{1}{{b - c}} = \frac{1}{a} + \frac{1}{c}\), then \(a,\;b,\;c\) are in

Answers Choices:

A.P.

G.P.

H.P.

In G.P. and H.P. both

Questions 28 of 50

Question:If \(a\) and \(b\) are two different positive real numbers, then which of the following relations is true

Answers Choices:

\(2\sqrt {ab} > (a + b)\)

\(2\sqrt {ab} < (a + b)\)

\(2\sqrt {ab} = (a + b)\)

None of these

Questions 29 of 50

Question:If \(a,\;b,\;c\) are in A.P., then \({3^a},\;{3^b},\;{3^c}\) shall be in

Answers Choices:

A.P.

G.P.

H.P.

None of these

Questions 30 of 50

Question:If the \({(m + 1)^{th}},\;{(n + 1)^{th}}\) and \({(r + 1)^{th}}\) terms of an A.P. are in G.P. and \(m,\;n,\;r\) are in H.P., then the value of the ratio of the common difference to the first term of the A.P. is

Answers Choices:

\( - \frac{2}{n}\)

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

\( - \frac{n}{2}\)

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

Questions 31 of 50

Question:An A.P., a G.P. and a H.P. have the same first and last terms and the same odd number of terms. The middle terms of the three series are in

Answers Choices:

A.P.

G.P.

H.P.

None of these

Questions 32 of 50

Question:If \(,a,\;b,\,c\) be in G.P. and \(a + x,\;b + x,\;c + x\) in H.P., then the value of \(x\) is (\(a,\;b,\;c\) are distinct numbers)

Answers Choices:

\(c\)

\(b\)

\(a\)

None of these

Questions 33 of 50

Question:If the ratio of H.M. and G.M. between two numbers \(a\) and \(b\) is \(4:5\), then the ratio of the two numbers will be

Answers Choices:

\(1:2\)

\(2:1\)

\(4:1\)

\(1:4\)

3 and 4 are correct

Questions 34 of 50

Question:If the A.M., G.M. and H.M. between two positive numbers \(a\) and \(b\) are equal, then

Answers Choices:

\(a = b\)

\(ab = 1\)

\(a > b\)

\(a < b\)

Questions 35 of 50

Question:If \(a,\;b,\;c\) are in A.P., then \({10^{ax + 10}},\;{10^{bx + 10}},\;{10^{cx + 10}}\) will be in

Answers Choices:

A.P.

G.P. only when \(x > 0\)

G.P. for all values of \(x\)

G.P. for \(x < 0\)

Questions 36 of 50

Question:The common difference of an A.P. whose first term is unity and whose second, tenth and thirty fourth terms are in G.P., is

Answers Choices:

\(\frac{1}{5}\)

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

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

\(\frac{1}{9}\)

Questions 37 of 50

Question:Let the positive numbers a, b, c, d be in A.P., then abc, abd acd, bcd are

Answers Choices:

Not in A.P./G.P./H.P.

In A.P.

In G.P.

In H.P.

Questions 38 of 50

Question:If in the equation \(a{x^2} + bx + c = 0,\) the sum of roots is equal to sum of square of their reciprocals, then \(\frac{c}{a},\frac{a}{b},\frac{b}{c}\) are in

Answers Choices:

A.P.

G.P.

H.P.

None of these

Questions 39 of 50

Question:The harmonic mean between two numbers is \(14\frac{2}{5}\)and the geometric mean 24 . The greater number them is

Answers Choices:

72

54

36

None of these

Questions 40 of 50

Question:When \(\frac{1}{a} + \frac{1}{c} + \frac{1}{{a - b}} + \frac{1}{{c - d}} = 0\) and \(b \ne a \ne c\), then \(a,\;b,\;c\) are

Answers Choices:

In H.P.

In G.P.

In A.P.

None of these

Questions 41 of 50

Question:If \({a^2},\,{b^2},\,{c^2}\) be in A.P., then \(\frac{a}{{b + c}},\,\frac{b}{{c + a}},\,\frac{c}{{a + b}}\) will be in

Answers Choices:

A.P.

G.P.

H.P.

None of these

Questions 42 of 50

Question:The sum of the first \(n\) terms of the series \(\frac{1}{2} + \frac{3}{4} + \frac{7}{8} + \frac{{15}}{{16}} + .........\) is

Answers Choices:

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

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

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

\({2^n} - 1\)

Questions 43 of 50

Question:The sum of \(1 + \frac{2}{5} + \frac{3}{{{5^2}}} + \frac{4}{{{5^3}}} + ...........\)upto \(n\) terms is

Answers Choices:

\(\frac{{25}}{{16}} - \frac{{4n + 5}}{{16 \times {5^{n - 1}}}}\)

\(\frac{3}{4} - \frac{{2n + 5}}{{16 \times {5^{n + 1}}}}\)

\(\frac{3}{7} - \frac{{3n + 5}}{{16 \times {5^{n - 1}}}}\)

\(\frac{1}{2} - \frac{{5n + 1}}{{3 \times {5^{n + 2}}}}\)

Questions 44 of 50

Question:\({2^{1/4}}{.4^{1/8}}{.8^{1/16}}{.16^{1/32}}..........\)is equal to

Answers Choices:

1

2

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

\(\frac{5}{2}\)

Questions 45 of 50

Question:The sum of the series \({1^2}.2 + {2^2}.3 + {3^2}.4 + ........\) to n terms is

Answers Choices:

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

\(\frac{{n(n + 1)(3{n^2} + 7n + 2)}}{{12}}\)

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

\(\frac{{n(n + 1)}}{{12}}[6n(n + 1) + 2(2n + 1)]\)

Questions 46 of 50

Question:The sum of the series \(1.2.3 + 2.3.4 + 3.4.5 + .......\) to n terms is

Answers Choices:

\(n(n + 1)(n + 2)\)

\((n + 1)(n + 2)(n + 3)\)

\(\frac{1}{4}n(n + 1)(n + 2)(n + 3)\)

\(\frac{1}{4}(n + 1)(n + 2)(n + 3)\)

Questions 47 of 50

Question:The sum of \({1^3} + {2^3} + {3^3} + {4^3} + ..... + {15^3}\),is [MP PET 2003]

Answers Choices:

22000

10,000

14,400

15,000

Questions 48 of 50

Question:The sum to \(n\) terms of the infinite series \({1.3^2} + {2.5^2} + {3.7^2} + ..........\infty \) is

Answers Choices:

\(\frac{n}{6}(n + 1)(6{n^2} + 14n + 7)\)

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

\(4{n^3} + 4{n^2} + n\)

None of these

Questions 49 of 50

Question:If the \({n^{th}}\) term of a series be \(3 + n\,(n - 1)\), then the sum of \(n\) terms of the series is

Answers Choices:

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

\(\frac{{{n^3} + 8n}}{3}\)

\(\frac{{{n^2} + 8n}}{5}\)

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

Questions 50 of 50

Question:The sum to \(n\) terms of \((2n - 1) + 2\,(2n - 3)\) \( + 3\,(2n - 5) + .....\) is

Answers Choices:

\((n + 1)\,(n + 2)\,(n + 3)/6\)

\(n\,(n + 1)\,(n + 2)/6\)

\(n\,(n + 1)\,(2n + 3)\,\)

\(n\,(n + 1)\,(2n + 1)/6\)