Sequences & Series · JEE Main previous year questions

A function f(x) is given by $f(x)=\frac{5^x}{5^x+5}$, then the sum of the series $$f\left(\frac{1}{20}\right)+f\left(\frac{2}{20}\right)+f\left(\frac{3}{20}\right)+\cdots +f\left(\frac{39}{20}\right)$$ is equal to :

1. A.$\frac{39}{2}$
2. B.$\frac{19}{2}$
3. C.$\frac{49}{2}$
4. D.$\frac{29}{2}$

The common difference of the A.P. $b_1,b_2,\dots ,b_m$ is 2 more than the common difference of A.P. $a_1,a_2,\dots ,a_n$. If $a_{40}=-159$, $a_{100}=-399$ and $b_{100}=a_{70}$, then $b_1$ is equal to :

1. A.127
2. B.81
3. C.$-127$
4. D.$-81$

If the first term of an A.P. is 3 and the sum of its first 25 terms is equal to the sum of its next 15 terms, then the common difference of this A.P. is :

1. A.$\frac{1}{4}$
2. B.$\frac{1}{5}$
3. C.$\frac{1}{7}$
4. D.$\frac{1}{6}$

If the sum of the series $20+19\frac{3}{5}+19\frac{1}{5}+18\frac{4}{5}+\dots$ upto nth term is 488 and the $n^{\mathrm{th}}$ term is negative, then :

1. A.$n=41$
2. B.$n=60$
3. C.$n^{\mathrm{th}}\text{ term is }-4$
4. D.$n^{\mathrm{th}}\text{ term is }-4\frac{2}{5}$

If m arithmetic means (A.Ms) and three geometric means (G.Ms) are inserted between 3 and 243 such that $4^{\mathrm{th}}$ A.M. is equal to $2^{\mathrm{nd}}$ G.M., then m is equal to _______.

The sum of the first three terms of a G.P. is S and their product is 27. Then all such S lie in :

1. A.$[-3,\infty )$
2. B.$(-\infty ,9]$
3. C.$(-\infty ,-9]\cup [-3,\infty )$
4. D.$(-\infty ,-3]\cup [9,\infty )$

If $|x|<1$, $|y|<1$ and $x\ne y$, then the sum to infinity of the following series $$(x+y)+(x^2+xy+y^2)+(x^3+x^{2}y+x{y}^2+y^3)+\dots$$

1. A.$\frac{x+y-xy}{(1+x)(1+y)}$
2. B.$\frac{x+y-xy}{(1-x)(1-y)}$
3. C.$\frac{x+y+xy}{(1+x)(1+y)}$
4. D.$\frac{x+y+xy}{(1-x)(1-y)}$

If the sum of first 11 terms of an A.P., $a_1,a_2,a_3,\dots$ is 0 $(a\ne 0)$, then the sum of the A.P., $a_1,a_3,a_5,\dots ,a_{23}$ is $k{a}_1$, where $k$ is equal to :

1. A.$\frac{121}{10}$
2. B.$-\frac{121}{10}$
3. C.$\frac{72}{5}$
4. D.$-\frac{72}{5}$

Let $S$ be the sum of the first 9 terms of the series : $\{x+ka\}+\{x^2+(k+2)a\}+\{x^3+(k+4)a\}+\{x^4+(k+6)a\}+\dots$ where $a\ne 0$ and $x\ne 1$. If $S=\frac{x^{10}-x+45a(x-1)}{x-1}$, then $k$ is equal to :

1. A.-3
2. B.1
3. C.-5
4. D.3

Let $a_n$ be the $n^{\mathrm{th}}$ term of a G.P. of positive terms. $$\sum _{n=1}^{100}{a}_{2n+1}=200\text{ and }\sum _{n=1}^{100}{a}_{2n}=100,$$ then ${\sum }_{n=1}^{200}{a}_n$ is equal to :

1. A.150
2. B.175
3. C.225
4. D.300

The number of terms common to the two A.P.'s 3, 7, 11, ....., 407 and 2, 9, 16, ....., 709 is ______.

Let $f:\mathbf{R}\to \mathbf{R}$ be such that for all $x\in \mathbf{R}$ $(2^{1+x}+2^{1-x})$, $f(x)$ and $(3^x+3^{-x})$ are in A.P., then the minimum value of $f(x)$ is

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

If the $10^{\text{th}}$ term of an A.P. is $\frac{1}{20}$ and its $20^{\text{th}}$ term is $\frac{1}{10}$, then the sum of its first 200 terms is

1. A.100
2. B.$100\frac{1}{2}$
3. C.$50\frac{1}{4}$
4. D.50

Five numbers are in A.P. whose sum is 25 and product is 2520. If one of these five numbers is $-\frac{1}{2}$ , then the greatest number amongst them is:

1. A.$\frac{21}{2}$
2. B.27
3. C.7
4. D.16

Let $a_1,a_2,a_3,\dots$ be a G.P. such that $a_1<0$, $a_1+a_2=4$ and $a_3+a_4=16$. If ${\sum }_{i=1}^9{a}_i=4\lambda$, then $\lambda$ is equal to:

1. A.171
2. B.-171
3. C.-513
4. D.$\frac{511}{3}$

If the sum of the first 40 terms of the series, 3 + 4 + 8 + 9 + 13 + 14 + 18 + 19 + ..... is (102)m, then m is equal to :

1. A.20
2. B.5
3. C.10
4. D.25

Let $S_n$ denote the sum of the first n terms of an A.P. If $S_4=16$ and $S_6=-48$, then $S_{10}$ is equal to :

1. A.-320
2. B.-380
3. C.-460
4. D.-210

For $x\in \mathbb{R}$, let [x] denote the greatest integer $\le x$, then the sum of the series $\left[-\frac{1}{3}\right]+\left[-\frac{1}{3}-\frac{1}{100}\right]+\left[-\frac{1}{3}-\frac{2}{100}\right]+\cdots +\left[-\frac{1}{3}-\frac{99}{100}\right]$ is :

1. A.-153
2. B.-135
3. C.-133
4. D.-131

If $\alpha$, $\beta$ and $\gamma$ are three consecutive terms of a non-constant G.P. such that the equations $\alpha x^2+2\beta x+\gamma =0$ and $x^2+x-1=0$ have a common root, then $\alpha (\beta +\gamma )$ is equal to :

1. A.$\alpha \gamma$
2. B.0
3. C.$\beta \gamma$
4. D.$\alpha \beta$

If $a_1,a_2,a_3,\dots$ are in A.P. such that $a_1+a_7+a_{16}=40$, then the sum of the first 15 terms of this A.P. is :

1. A.120
2. B.200
3. C.150
4. D.280

If $a_1,a_2,a_3,\dots ,a_n$ are in A.P. and $a_1+a_4+a_7+\cdots +a_{16}=114$, then $a_1+a_6+a_{11}+a_{16}$ is equal to :

1. A.38
2. B.98
3. C.76
4. D.64

Let $a$, b and c be in G.P. with common ratio r, where $a\ne 0$ and $0<r\le \frac{1}{2}$. If 3a, 7b and 15c are the first three terms of an A.P., then the $4^{\mathrm{th}}$ term of this A.P. is :

1. A.$a$
2. B.$\frac{7}{3}a$
3. C.5a
4. D.$\frac{2}{3}a$

Let $a_1,a_2,a_3,\dots$ be an A.P. with $a_6=2$. Then the common difference of this A.P., which maximises the product $a_1{a}_4{a}_5$, is :

1. A.$\frac{3}{2}$
2. B.$\frac{6}{5}$
3. C.$\frac{8}{5}$
4. D.$\frac{2}{3}$

The angles A, B and C of a triangle ABC are in A.P. and $a:b=1:\sqrt{3}$. If c = 4 cm, then the area (in sq. cm) of this triangle is :

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

Let the sum of the first n terms of a non-constant A.P., $a_1,a_2,a_3,\dots$ be $50n+\frac{n(n-7)}{2}A$, where A is a constant. If d is the common difference of this A.P., then the ordered pair $(d,a_{50})$ is equal to

1. A.(A, 50+45A)
2. B.(50, 50+45A)
3. C.(A, 50+46A)
4. D.(50, 50+46A)