Sequences & Series · JEE Main previous year questions

Let $x_1,x_2,x_3,\dots ,x_{20}$ be in geometric progression with $x_1=3$ and the common ratio $\frac{1}{2}$. A new data is constructed replacing each $x_i$ by $(x_i-i{)}^2$. If $\bar{x}$ is the mean of new data, then the greatest integer less than or equal to $\bar{x}$ is ____.

Suppose $a_1,a_2,\dots ,a_n,\dots$ be an arithmetic progression of natural numbers. If the ratio of the sum of first five terms to the sum of first nine terms of the progression is $5:17$ and $110<a_{15}<120$, then the sum of the first ten terms of the progression is equal to

1. A.290
2. B.380
3. C.460
4. D.510

Let $A(1,1),B(-4,3),C(-2,-5)$ be vertices of a triangle $ABC$, $P$ be a point on side $BC$, and ${\Delta }_1$ and ${\Delta }_2$ be the areas of triangles $APB$ and $ABC$, respectively. If ${\Delta }_1:{\Delta }_2=4:7$, then the area enclosed by the lines $AP,AC$ and the $x$-axis is :

1. A.$\frac{1}{4}$
2. B.$\frac{3}{4}$
3. C.$\frac{1}{2}$
4. D.1

Let the sum of an infinite G.P., whose first term is a and the common ratio is r, be $5$ . Let the sum of its first five terms be $\frac{98}{25}$ . Then the sum of the first $21$ terms of an AP, whose first term is $10\mathrm{ar}$, ${\mathrm{n}}^{\mathrm{th}}$ term is ${\mathrm{a}}_{\mathrm{n}}$ and the common difference is $10{\mathrm{ar}}^2$, is equal to :

1. A.$21{\mathrm{a}}_{11}$
2. B.$22{\mathrm{a}}_{11}$
3. C.$15{\mathrm{a}}_{16}$
4. D.$14{\mathrm{a}}_{16}$

Consider two G.Ps. $2,2^2,2^3,\dots$ and $4,4^2,4^3,\dots$ of 60 and n terms respectively. If the geometric mean of all the $60+n$ terms is $(2{)}^{\frac{225}{8}}$, then ${\sum }_{k=1}^{n}k(n-k)$ is equal to :

1. A.560
2. B.1540
3. C.1330
4. D.2600

The series of positive multiples of 3 is divided into sets : $\{3\}$, $\{6,9,12\}$, $\{15,18,21,24,27\}$, \dots . Then the sum of the elements in the $11^{\mathrm{th}}$ set is equal to ____ .

Let for $f(x)=a_0{x}^2+a_{1}x+a_2$, $f^{\prime }(0)=1$ and $f^{\prime }(1)=0$. If ${\mathrm{a}}_0$, ${\mathrm{a}}_1$, ${\mathrm{a}}_2$ are in an arithmetico-geometric progression, whose corresponding A.P. has common difference 1 and corresponding G.P. has common ratio 2, then $f(4)$ is equal to ____ .

If n arithmetic means are inserted between a and 100 such that the ratio of the first mean to the last mean is 1 : 7 and a + n = 33, then the value of n is :

1. A.21
2. B.22
3. C.23
4. D.24

Let for $n=1,2,\dots ,50$, $S_n$ be the sum of the infinite geometric progression whose first term is $n^2$ and whose common ratio is $\frac{1}{(n+1{)}^2}$. Then the value of $\frac{1}{26}+{\sum }_{n=1}^{50}\left(S_n+\frac{2}{n+1}-n-1\right)$ is equal to ________.

If the sum of the first ten terms of the series $$\frac{1}{5}+\frac{2}{65}+\frac{3}{325}+\frac{4}{1025}+\frac{5}{2501}+\dots$$ is $\frac{m}{n}$, where $m$ and $n$ are co-prime numbers, then $m+n$ is equal to ____.

If $a_1,a_2,a_3\dots \dots$ and $b_1,b_2,b_3\dots \dots$ are A.P., and $a_1=2,a_{10}=3,a_1{b}_1=1=a_{10}{b}_{10}$, then $a_4{b}_4$ is equal to -

1. A.$\frac{35}{27}$
2. B.1
3. C.$\frac{27}{28}$
4. D.$\frac{28}{27}$

If $a_1(>0),a_2,a_3,a_4,a_5$ are in a G.P., $a_2+a_4=2a_3+1$ and $3a_2+a_3=2a_4$, then $a_2+a_4+2a_5$ is equal to ________.

The greatest integer less than or equal to the sum of first 100 terms of the sequence $\frac{1}{3}$, $\frac{5}{9}$, $\frac{19}{27}$, $\frac{65}{81}$, $\dots$ is equal to $\underline{}$.

Let $a_1,a_2,\dots ,a_{21}$ be an AP such that ${\sum }_{n=1}^{20}\frac{1}{a_n{a}_{n+1}}=\frac{4}{9}$. If the sum of this AP is 189, then $a_6{a}_{16}$ is equal to :

1. A.57
2. B.72
3. C.48
4. D.36

Let $a_1,a_2,\dots ,a_{10}$ be an AP with common difference $-3$ and $b_1,b_2,\dots ,b_{10}$ be a GP with common ratio 2. Let $c_k=a_k+b_k$, $k=1,2,\dots ,10$. If $c_2=12$ and $c_3=13$, then ${\sum }_{k=1}^{10}{c}_k$ is equal to ________.

If $\tan \left(\frac{\pi }{9}\right),x,\tan \left(\frac{7\pi }{18}\right)$ are in arithmetic progression and $\tan \left(\frac{\pi }{9}\right),y,\tan \left(\frac{5\pi }{18}\right)$ are also in arithmetic progression, then $|x-2y|$ is equal to :

1. A.4
2. B.3
3. C.0
4. D.1

Let $S_n$ be the sum of the first n terms of an arithmetic progression. If $S_{3n}=3S_{2n}$, then the value of $\frac{S_{4n}}{S_{2n}}$ is :

1. A.6
2. B.4
3. C.2
4. D.8

Let $S_n$ denote the sum of first n-terms of an arithmetic progression. If $S_{10}=530$, $S_5=140$, then $S_{20}-S_6$ is equal to:

1. A.$1862$
2. B.$1842$
3. C.$1852$
4. D.$1872$

Let a, b, c, d in arithmetic progression with common difference $\lambda$. If $$\left|\begin{array}{ccc}x+a-c& x+b& x+a\\ x-1& x+c& x+b\\ x-b+d& x+d& x+c\end{array}\right|=2,$$ then value of ${\lambda }^2$ is equal to ________.

Let ${\mathrm{S}}_1$ be the sum of first 2n terms of an arithmetic progression. Let ${\mathrm{S}}_2$ be the sum of first 4n terms of the same arithmetic progression. If $({\mathrm{S}}_2-{\mathrm{S}}_1)$ is 1000, then the sum of the first 6n terms of the arithmetic progression is equal to :

1. A.7000
2. B.1000
3. C.3000
4. D.5000

Consider an arithmetic series and a geometric series having four initial terms from the set {11, 8, 21, 16, 26, 32, 4}. If the last terms of these series are the maximum possible four digit numbers, then the number of common terms in these two series is equal to ___________.

In an increasing geometric series, the sum of the second and the sixth term is $\frac{25}{2}$ and the product of the third and fifth term is 25. Then, the sum of $4^{\text{th}},6^{\text{th}}$ and $8^{\text{th}}$ terms is equal to :

1. A.30
2. B.32
3. C.26
4. D.35

The sum of the infinite series $1+\frac{2}{3}+\frac{7}{3^2}+\frac{12}{3^3}+\frac{17}{3^4}+\frac{22}{3^5}+\dots \dots$ is equal to :

1. A.$\frac{9}{4}$
2. B.$\frac{13}{4}$
3. C.$\frac{15}{4}$
4. D.$\frac{11}{4}$

The sum of the series ${\sum }_{n=1}^{\infty }\frac{n^2+6n+10}{(2n+1)!}$ is equal to :

1. A.$\frac{41}{8}e+\frac{19}{8}{e}^{-1}-10$
2. B.$\frac{41}{8}e-\frac{19}{8}{e}^{-1}-10$
3. C.$\frac{41}{8}e+\frac{19}{8}{e}^{-1}+10$
4. D.$-\frac{41}{8}e+\frac{19}{8}{e}^{-1}-10$

If the arithmetic mean and geometric mean of the $p^{\text{th}}$ and $q^{\text{th}}$ terms of the sequence $-16,8,-4,2,\dots \dots$ satisfy the equation $4x^2-9x+5=0$, then $p+q$ is equal to _________.