IISER Mathematics - Sequences and Series
Exam Duration: 45 Mins Total Questions : 30
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 a, b, c, d and p are distinct natural numbers such that \((a^2+b^2+c^2)P^2-(ab+bc+cd)p+(b^2+c^2+d^2)\le 0\) then
- (a)
a, b, c, d are in A.P.
- (b)
a, b, c, d are in G.P.
- (c)
a, b, c, d are in H.P.
- (d)
ab = cd
The sum of the series \(1+\frac { 1+2 }{ 2! } +\frac { 1+2+3 }{ 3! } +\frac { 1+2+3+4 }{ 4! } +...\infty \) is
- (a)
e
- (b)
\(3e\over 2\)
- (c)
2e
- (d)
\(5e\over 2\)
The value \(1+\frac { { 2 }^{ 3 } }{ 2! } +\frac { { 3 }^{ 3 } }{ 3! } +\frac { { 4 }^{ 3 } }{ 4! } +...+\infty \) is
- (a)
e3
- (b)
e2
- (c)
3e
- (d)
5e
\({ S }_{ n }=\frac { { 1 }^{ 2 }.2 }{ 1! } +\frac { { 2 }^{ 2 }.3 }{ 2! } +\frac { { 3 }^{ 2 }.4 }{ 3! } ...\infty \); then Sn has the value
- (a)
3e
- (b)
5e
- (c)
7e
- (d)
9e
If \({ 5 }^{ 1+x }+{ 5 }^{ 1-x },\frac { a }{ 2 } \) and \({ 25 }^{ x }+{ 25 }^{ -x }\) are three consecutive terms of an AP, then the values of a are given by
- (a)
\(a\ge 12\)
- (b)
\(a= 13\)
- (c)
\(a\ge 13\)
- (d)
\(a\le 12\)
If x, Y and z are positive integers, then value of expression ( X + Y) (Y + Z) (Z + X) is
- (a)
= 8 xyz
- (b)
> 8 xyz
- (c)
< 8 xyz
- (d)
= 4 xyz
The minimum value of the expression \({ 3 }^{ X }+{ 3 }^{ 1-X },X\epsilon R,\) is
- (a)
0
- (b)
\(\frac{1}{3}\)
- (c)
3
- (d)
\(2\sqrt{3}\)
If \({ log }_{ a }X,{ log }_{ b }X\) and \({ log }_{ c }X\) are in HP, then a,b and c are in
- (a)
AP
- (b)
HP
- (c)
GP
- (d)
None of the above
The sum upto n terms of the series \(\tan ^{ -1 }{ \frac { 1 }{ 2 } } +\tan ^{ -1 }{ \frac { 2 }{ 9 } } +\tan ^{ -1 }{ \frac { 1 }{ 8 } } +\tan ^{ -1 }{ \frac { 2 }{ 25 } } +\tan ^{ -1 }{ \frac { 1 }{ 18 } } +...\) is
- (a)
\(\tan ^{ -1 }{ \left( \frac { 1 }{ 3 } \right) } \)
- (b)
\(\frac { \pi }{ 4 } \)
- (c)
\(\tan ^{ -1 }{ \frac { 1 }{ 2 } } \)
- (d)
\(\cot ^{ -1 }{ 2 } \)
Let \({ a }_{ n }\) be the nth term of an AP. If \(\sum _{ r=1 }^{ 100 }{ { a }_{ 2r } } =\alpha \quad and\quad \sum _{ r=1 }^{ 100 }{ { a }_{ 2r-1 } } =\beta \) then the common difference of the AP is
- (a)
\(\frac { \alpha -\beta }{ 200 } \)
- (b)
\(\alpha -\beta \)
- (c)
\(\frac { \alpha -\beta }{ 100 } \)
- (d)
\(\beta -\alpha \)
The 1025th term in the sequence 1, 22, 4444, 88888888, ...is
- (a)
29
- (b)
210
- (c)
211
- (d)
212
If a, b, c are in AP, then the equation (a - b) x2 + (c - a) x + (b - c) = 0 has two roots which are
- (a)
rational and equal
- (b)
rational and distinct
- (c)
irrational conjugates
- (d)
complex conjugates
If x, (2x + 2), (3x + 3),... are in GP, then the next term of this sequence is
- (a)
27
- (b)
- 27
- (c)
13.5
- (d)
-13.5
If log2 (a+b) +log2(c+d)\(\ge \)4. Then the minimum value of the expression a + b + c + d is
- (a)
2
- (b)
4
- (c)
8
- (d)
none of these
If p, q, r are three positive real numbers are in AP, then the roots of the quadratic equation px2 + qx + r = 0 are all real for
- (a)
\(\left| \frac { r }{ p } -7 \right| \ge 4\sqrt { 3 } \)
- (b)
\(\left| \frac { P }{ r } -7 \right| <4\sqrt { 3 } \)
- (c)
all p and r
- (d)
no p and r
The sum of the products of ten numbers ± 1, ± 2, ± 3, ± 4, ± 5 taking two at a time is
- (a)
- 55
- (b)
55
- (c)
165
- (d)
-165
Let a, b, c be three positive prime numbers. The progression in which \(\sqrt { a } ,\sqrt { b } ,\sqrt { c } \) can be three terms (not necessarily consecutive) is
- (a)
AP
- (b)
GP
- (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.
\(\alpha\) 2 lies in
- (a)
\(\left( \frac { 1 }{ 3 } ,1 \right) \)
- (b)
(1, 2)
- (c)
\(\left( \frac { 1 }{ 3 } ,3 \right) \)
- (d)
\(\left( \frac { 1 }{ 3 } ,1 \right) \cup (1,3)\)
The sum of the squares of three distinct real numbers which are in strictly increasing GP is S 2. If their sum is \(\alpha\) S.
If S = 10\(\sqrt { 3 } \) , then the greatest value of the middle term is
- (a)
5
- (b)
5\(\sqrt { 3 } \)
- (c)
10
- (d)
10\(\sqrt { 3 } \)
We are giving the concept of arithmetic mean of m th power. Let a1,a2, a3,.... a n be n be n positive real numbers (not all equal) and let m be real number.
Then \(\frac { { a }_{ 1 }^{ m }+{ a }_{ 2 }^{ m }+{ a }_{ 3 }^{ m }+.....+{ a }_{ n }^{ m } }{ n } >{ \left( \frac { { a }_{ 1 }+{ a }_{ 2 }+{ a }_{ 3 }+....+{ a }_{ n } }{ n } \right) }^{ m };if\quad m\in R\sim [0,\quad 1]\)
However if \(m\in (0,1),\quad then\quad \frac { { a }_{ 1 }^{ m }+{ a }_{ 2 }^{ m }+{ a }_{ 3 }^{ m }+.....+{ a }_{ n }^{ m } }{ n } <{ \left( \frac { { a }_{ 1 }+{ a }_{ 2 }+{ a }_{ 3 }+...+{ a }_{ n } }{ n } \right) }^{ m }\)
Obviously if m\(\in \){0, 1}, then \(\frac { { a }_{ 1 }^{ m }+{ a }_{ 2 }^{ m }+{ a }_{ 3 }^{ m }+.....+{ a }_{ n }^{ m } }{ n } ={ \left( \frac { { a }_{ 1 }+{ a }_{ 2 }+{ a }_{ 3 }+...+{ a }_{ n } }{ n } \right) }^{ m }\)
If a, b, c, d be positive and not all equal to one another such that a+b+c+d = 4/3, then the minimum value of \(\sqrt { \left( \frac { 1 }{ b+c+d } \right) } +\sqrt { \left( \frac { 1 }{ a+b+c } \right) } \) is
- (a)
2
- (b)
4
- (c)
6
- (d)
8
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)
\({-1\over2},0,{1\over2}\)
Let the sequence an be defined as follows :
a1=1,an=an-1+2 for n \(\ge\)2.
Write the corresponding series.
- (a)
1+3+5+7+9+.......
- (b)
1+5+9+13+17+.....
- (c)
1+4+7+10+13+.........
- (d)
1+6+11+16+21+......
In an A.P., the pth term is q and the (p+q)th term is 0. Then the qth term is
- (a)
-p
- (b)
p
- (c)
p+q
- (d)
p-q
Find the sum to n terms of the series: \({1\over1.3}+{1\over3.5}+{1\over5.7}+...\)
- (a)
\({n\over(2n-1)}\)
- (b)
\({n\over(2n+1)}\)
- (c)
n+1
- (d)
n-1
The sum of the series1.n+2.(n-1)+3.(n-2)+...+n.1=
- (a)
\({n(n+1)(2n+1)\over6}\)
- (b)
\({n(n+1)(n+2)\over6}\)
- (c)
\({n(n+1)(n+2)\over3}\)
- (d)
\({n(n+1)(2n+1)\over3}\)
The sum to 20terms of the series 1x32+2x52+3x72+...... is
- (a)
18800
- (b)
188010
- (c)
188020
- (d)
188090
\({1^3\over1}+{1^3+2^3+3^3\over 1+3+5}+....\)to 16 terms=
- (a)
420
- (b)
416
- (c)
436
- (d)
446
113-103+93-83+73-63+53-43+33-23+13=
- (a)
756
- (b)
724
- (c)
648
- (d)
812
Which of the following statements is/are true?
Statement-I: If\(\theta_1,\theta_2,\theta_3,....\theta_n\) are in A.P. whose common difference is d, then sec\(\theta_1\)sec \(\theta_2\) + sec\(\theta_2\) sec\(\theta_3\) + .....+ sec\(\theta_{n-1}\) sec \(\theta_n\) = \({tan \theta_n-tan \theta_1\over sin d}\)
Statement-II: If pth, qth and rth terms of an A.P. and G.P. are a, b and c respectively, then ab-c, bc-a, ca-b = 1.
- (a)
Only Statement-I
- (b)
Only Staternent-Il
- (c)
Both Statement-I and Staternent-Il
- (d)
Neither Staternent-I nor Staternent-Il