Mathematics - Sequences and Series
Exam Duration: 45 Mins Total Questions : 30
Let the positive numbers a, b, c, d be in A.P. Then abc, abd, bcd are
- (a)
not in A.P./G.P./H.P.
- (b)
in A.P.
- (c)
in G.P.
- (d)
in H.P.
The 4th term of an A.P. is 11 and the 11th terms is 25. The common difference of the series is
- (a)
2
- (b)
3
- (c)
-3
- (d)
1
The arithmetic mean between two numbers is A and the geometric means is G. Then these numbers are
- (a)
\(A\pm \sqrt { { A }^{ 2 }-{ G }^{ 2 } } \)
- (b)
\(\sqrt { { A }^{ 2 }-{ G }^{ 2 } } \pm A\)
- (c)
\(\frac { 1 }{ 2 } \left( \sqrt { { A }^{ 2 }-{ +G }^{ 2 } } +A \right) \)
- (d)
NONE OF THESE
The sum of the series \(2+\frac { 3 }{ 2 } +1+\frac { 5 }{ 8 } +...\infty \) is
- (a)
5
- (b)
6
- (c)
7
- (d)
NONE OF THESE
If a, b, c are in G.P., then the equations \(ax^2+2bx+c = 0\) and \(dx^2+2ex+f=0\) have a common root \(\frac { d }{ a } ,\frac { e }{ b } ,\frac { f }{ c } \) are in
- (a)
A.P.
- (b)
G.P.
- (c)
H.P.
- (d)
NONE OF THESE
Sum of products of first n natural numbers, taken two at a time, is
- (a)
\({1\over 6}n(n+1)(n+2)(n+5)\)
- (b)
\({1\over 24} (n-1)(n)(n+1)(3n+2)\)
- (c)
\({1\over 48} (n-2)(n-1)n^2\)
- (d)
NONE OF THESE
If a1, a2, a3,..., an are non-zero real numbers such that \(({ a }_{ 1 }^{ 2 }\quad +\quad { a }_{ 2 }^{ 2 }\quad +\quad { a }_{ 3 }^{ 2 }\quad +...+\quad { a }_{ n-1 }^{ 2 })\quad ({ a }_{ 2 }^{ 2 }\quad +\quad { a }_{ 3 }^{ 2 }\quad +...+{ a }_{ n }^{ 2 })\quad \le \quad (a_1a_2 \ +\ a_2a_3\ + ...a_{n-1}a_n)^2\)
then \(a_1,a_2,a_3,...a_n\) are in
- (a)
AP
- (b)
G.P.
- (c)
H.P.
- (d)
NONE OF THESE
If x, y, z (>0) then minimum value of xlog y - log z + ylog z - log x + zlog x-log y is
- (a)
3
- (b)
1
- (c)
0
- (d)
NONE OF THESE
Sum of the following series equals \(1+\frac { 1+2 }{ 2! } +\frac { 1+2+{ 2 }^{ 2 } }{ 3! } +\frac { 1+2+{ 2 }^{ 2 }+{ 2 }^{ 3 } }{ 4! } +...\infty \)
- (a)
e2-1
- (b)
e2+1
- (c)
e2-e
- (d)
e2+e
Value of \({ log }_{ e }\frac { (1+{ x }^{ 3 }) }{ (1-x+{ x }^{ 2 }) } \), is
- (a)
loge (1-x)
- (b)
loge(1+x)
- (c)
-loge (1+x)
- (d)
-loge (1-x)
Let S be the sum, P be the product and R be, the sum of the reciprocals of 3 terms of a GP. Then, P2 R3 : S3 is equal to
- (a)
1 : 1
- (b)
(common ratio)n : 1
- (c)
(first term)2 : (common ratio) 2
- (d)
None of the above
A number consists of three digits in GP. The sum of the right hand and left hand digits exceeds twice the middle digit by 1 and the sum of the left hand and middle digits is two third of the sum of the middle and right hand digits. Then, the number is
- (a)
468
- (b)
469
- (c)
964
- (d)
649
The series is given as \(\sum _{ r=1 }^{ n }{ \left\{ \frac { r }{ \left( r+1 \right) ! } \right\} } \), then the sum of series is
- (a)
\(2+\frac { 2 }{ n! } \)
- (b)
\(1-\frac { 1 }{ \left( n+1 \right) ! } \)
- (c)
\(2+\frac { 1 }{ \left( n+1 \right) ! } \)
- (d)
\(1+\frac { 1 }{ \left( n+1 \right) ! } \)
Three positive numbers form an increasing GP. If the middle term in this GP is doubled, then the new numbers are in AP. The common ratio of the GP is
- (a)
\(\sqrt { 2 } +\sqrt { 3 } \)
- (b)
\(3 +\sqrt { 2 } \)
- (c)
\(2 -\sqrt { 3 } \)
- (d)
\(2 +\sqrt { 3 } \)
A man saves Rs. 200 in each of the first three months of his service. In each of the subsequent months, his saving of immediately previous month. His total saving from the start of service will be Rs. 11040 after
- (a)
19 months
- (b)
20 months
- (c)
21 months
- (d)
18 months
Let \({ a }_{ 1 },{ a }_{ 2 },{ a }_{ 3 }...\) be terms of an AP. If \(\frac { { a }_{ 1 }+{ a }_{ 2 }+...+{ a }_{ p } }{ { a }_{ 1 }+{ a }_{ 2 }+...+{ a }_{ q } } =\frac { { p }^{ 2 } }{ { q }^{ 2 } } ,p\neq q,\) then \(\frac { { a }_{ 6 } }{ { a }_{ 21 } } \) equals
- (a)
\(\frac{7}{2}\)
- (b)
\(\frac{2}{7}\)
- (c)
\(\frac{11}{41}\)
- (d)
\(\frac{41}{11}\)
Number of identical terms in the sequence 2, 5, 8, 11, .. upto 100 terms and 3,5,7,9, 11, ... upto 100 terms are
- (a)
17
- (b)
33
- (c)
50
- (d)
147
The coefficient of x49 in the product (x - 1) (x - 3)....(x - 99) is
- (a)
-992
- (b)
1
- (c)
-2500
- (d)
none of these
If a1 = 0 and a1,a2,a3,......,an are real numbers such that \(\left| { a }_{ i } \right| =\left| { a }_{ i-1 }+1 \right| \) for all i, then the arithmetic mean of the numbers a1, a2 ,....,an has value x where
- (a)
x < -1
- (b)
\(x<-\frac { 1 }{ 2 } \)
- (c)
\(x\ge -\frac { 1 }{ 2 } \)
- (d)
\(x=-\frac { 1 }{ 2 } \)
If 1n (a+c), 1n(c-a),1n(a-2b+c) are in AP, then
- (a)
a, b, c are in AP
- (b)
a2, b2, c2 are in AP
- (c)
a, b, c are in GP
- (d)
a, b, c are in HP
Let a, b, c be positive real numbers, such that bx2+ \(\left( \sqrt { { (a+c) }^{ 2 }+{ 4b }^{ 2 } } \right) x+(a+c)\ge 0,\quad \forall \quad x\in R,\) then a, b, c are in
- (a)
GP
- (b)
AP
- (c)
HP
- (d)
none of these
The sum of the squares of three distinct real numbers which are in strictly increasing GP is S2. If their sum is \(\alpha\) S.
If r = 2, then the value of (\(\alpha\)2) is (where (.) denotes the least integer function and r is common ratio of GP)
- (a)
0
- (b)
1
- (c)
2
- (d)
3
Write the first three terms of the sequence whose general term is \(a_n={n-3\over4}\)
- (a)
\({-1\over2},{-1\over4},0\)
- (b)
\({-1\over2},{-1\over3},{-1\over4}\)
- (c)
-1,-2,-3
- (d)
\({-1\over2},0,{1\over2}\)
How many terms of the G.P.3,\({3\over2},{3\over4},....\) are needed to give the sum \({3069\over512}?\)
- (a)
9
- (b)
12
- (c)
8
- (d)
10
In a G.P. of even number of terms, the sum of all terms is 5 times the sum of the odd terms. The common ratio of the G.P. is
- (a)
\({-4\over 5}\)
- (b)
\({1\over5}\)
- (c)
4
- (d)
-5
The ratio of the sum of first 3 terms to the sum of first 6terms of a G.P. is 125:152. The common ratio of the G.P., is
- (a)
\(2\over3\)
- (b)
\(3\over4\)
- (c)
\(3\over5\)
- (d)
\(2\over5\)
The A.M. of the roots of a quadratic equation is A and G.M. of its roots is G. The quadratic equation is
- (a)
x2+Ax+G2=0
- (b)
x2+2Ax+G2=0
- (c)
x2-Ax+G2=0
- (d)
x2-2Ax+G2=0
Let Sn denote the sum of the first n terms of an A.P. If S2n = 3Sn then S3n : Sn is equal to
- (a)
4
- (b)
6
- (c)
8
- (d)
10
Statement- I : The ratio of sum of m terms to the sum of n terms of an A.P is m2 : n2. If Tk is the kth term, thenT5/T2 = 3.
Statement-II : For nth term, tn = a + (n - 1)d, where 'a' is first term and 'd' is common difference.
- (a)
If both Statement-I and Statement-II are true and Statement-II is the correct explanation of Statement -1.
- (b)
If both Statement-I and Statement-II are true but Statement-II is not the correct explanation of Statement -1.
- (c)
If Statement-I is true but Statement-II is false.
- (d)
If Statement-I is false and Statement-II is true.
Let us define the sum of cubic numbers is \(\sum n^{3}=[{n(n+1)\over2}]^2\)
Statement-I : Sum of the series 13 - 23 + 33- 43 + ... + 113 = 756
Statement-II: For any odd integer 1 n \(\ge\) 1, n3 - (n -1)3 + ... + (-1)n-1 13= \(1\over4\)(2n -1)(n + 1)2.
- (a)
If both Statement-I and Statement-II are true and Statement-II is the correct explanation of Statement -1.
- (b)
If both Statement-I and Statement-II are true but Statement-II is not the correct explanation of Statement -1.
- (c)
If Statement-I is true but Statement-II is false.
- (d)
If Statement-I is false and Statement-II is true.