Sequences & Series · JEE Main previous year questions

The sum of the first ten terms of an A.P. is $160$ and the sum of the first two terms of a G.P. is $8$. If the first term of the A.P. is equal to the common ratio of the G.P. and the first term of the G.P. is equal to common difference of the A.P., then the sum of all possible values of the first term of the G.P. is:

1. A.$\frac{34}{9}$
2. B.$\frac{34}{13}$
3. C.$\frac{32}{9}$
4. D.$\frac{32}{13}$

If $(1-x^3{)}^{10}={\sum }_{r=0}^{10}{a}_r{x}^r(1-x{)}^{30-2r}$, then $\frac{9a_9}{a_{10}}$ is equal to ____ .

Let the sum of the first $n$ terms of an A.P. be $3n^2+5n$. Then the sum of squares of the first $10$ terms of the A.P. is:

1. A.10220
2. B.12860
3. C.15220
4. D.19780

If the sum of the first 10 terms of the series $$\frac{1}{1+1^4\times 4}+\frac{2}{1+2^4\times 4}+\frac{3}{1+3^4\times 4}+\frac{4}{1+4^4\times 4}+\dots$$ is $\frac{m}{n}$, $\gcd (m,n)=1$, then $m+n$ is equal to :

1. A.$256$
2. B.$264$
3. C.$276$
4. D.$284$

Let $A_1,A_2,A_3,\dots ,A_{39}$ be $39$ arithmetic means between the numbers $59$ and $159$. Then the mean of $A_{25},A_{28},A_{31}$ and $A_{36}$ is equal to :

1. A.$129$
2. B.$136$
3. C.$131.50$
4. D.$134$

The first term of an A.P. of 30 non-negative terms is $\frac{10}{3}$. If the sum of this A.P. is the cube of its last term, then its common difference is:

1. A.$\frac{5}{87}$
2. B.$\frac{25}{83}$
3. C.$\frac{15}{29}$
4. D.$\frac{5}{29}$

Let $\mathrm{A}$ be the set of first $101$ terms of an A.P., whose first term is $1$ and the common difference is $5$ and let $\mathrm{B}$ be the set of first $71$ terms of an A.P., whose first term is $9$ and the common difference is $7$. Then the number of elements in $\mathrm{A}\cap \mathrm{B}$, which are divisible by $3$, is :

1. A.4
2. B.5
3. C.6
4. D.7

Gas 'A' undergoes change from state 'X' to state 'Y'. In this process, the heat absorbed and work done by the gas is 10 J and 18 J respectively. Now gas is brought back to state 'X' by another process during which 6 J of heat is evolved. In the reverse process of 'Y' to 'X',

1. A.18 J of the work is done by the gas 'A'.
2. B.2 J of the work is done by the gas 'A'.
3. C.12 J of the work is done on the gas 'A' by the surrounding.
4. D.14 J of the work is done on the gas 'A' by the surrounding.

The common difference of the A.P.: $a_1,a_2,\dots ,a_m$ is $13$ more than the common difference of the A.P.: $b_1,b_2,\dots ,b_n$. If $b_{31}=-277,b_{43}=-385$ and $a_{78}=327$, then $a_1$ is equal to

1. A.21
2. B.19
3. C.24
4. D.16

Let the arithmetic mean of $\frac{1}{a}$ and $\frac{1}{b}$ be $\frac{5}{16},a>2$. If $\alpha$ is such that $a,4,\alpha ,b$ are in A.P., then the equation $\alpha x^2-ax+2(\alpha -2b)=0$ has :

1. A.one root in $(1,4)$ and another in $(-2,0)$
2. B.one root in $(0,2)$ and another in $(-4,-2)$
3. C.both roots in the interval $(-2,0)$
4. D.complex roots of magnitude less than 2

Let $729,81,9,1,\dots$ be a sequence and ${\text{P}}_n$ denote the product of the first $n$ terms of this sequence. If $2{\sum }_{n=1}^{40}({\text{P}}_n{)}^{\frac{1}{n}}=\frac{3^{\alpha }-1}{3^{\beta }}$ and $\gcd (\alpha ,\beta )=1$, then $\alpha +\beta$ is equal to

1. A.73
2. B.74
3. C.75
4. D.76

Consider an A.P.: $a_1,a_2,\dots ,a_n;a_1>0$. If $a_2-a_1=\frac{-3}{4}$, $a_n=\frac{1}{4}{a}_1$, and ${\sum }_{i=1}^n{a}_i=\frac{525}{2}$, then ${\sum }_{i=1}^{17}{a}_i$ is equal to

1. A.238
2. B.136
3. C.476
4. D.952

The sum of all possible values of $n\in \mathbf{N}$, so that the coefficients of $x$, $x^2$ and $x^3$ in the expansion of $(1+x^2{)}^2(1+x{)}^n$, are in arithmetic progression is :

1. A.12
2. B.9
3. C.3
4. D.7

If the sum of the first four terms of an A.P. is 6 and the sum of its first six terms is 4 , then the sum of its first twelve terms is

1. A.-26
2. B.-20
3. C.-24
4. D.-22

Let $a_1,\frac{a_2}{2},\frac{a_3}{2^2},\dots ,\frac{a_{10}}{2^9}$ be a G.P. of common ratio $\frac{1}{\sqrt{2}}$. If $a_1+a_2+\dots +a_{10}=62$, then $a_1$ is equal to:

1. A.$\sqrt{2}-1$
2. B.$2(\sqrt{2}-1)$
3. C.$2-\sqrt{2}$
4. D.$2(2-\sqrt{2})$

Let $A=\{1,6,11,16,\dots \}$ and $B=\{9,16,23,30,\dots \}$ be the sets consisting of the first $2025$ terms of two arithmetic progressions. Then $n(A\cup B)$ is

1. A.$3814$
2. B.$4003$
3. C.$4027$
4. D.$3761$

If the sum of the first 20 terms of the series $\frac{4\cdot 1}{4+3\cdot 1^2+1^4}+\frac{4\cdot 2}{4+3\cdot 2^2+2^4}+\frac{4\cdot 3}{4+3\cdot 3^2+3^4}+\frac{4\cdot 4}{4+3\cdot 4^2+4^4}+\dots$ is $\frac{m}{n}$, where $m$ and $n$ are coprime, then $m+n$ is equal to :

1. A.423
2. B.421
3. C.422
4. D.420

The sum of the infinite series ${\cot }^{-1}\left(\frac{7}{4}\right)+{\cot }^{-1}\left(\frac{19}{4}\right)+{\cot }^{-1}\left(\frac{39}{4}\right)+{\cot }^{-1}\left(\frac{67}{4}\right)+\dots$ is :

1. A.$\frac{\pi }{2}+{\cot }^{-1}\left(\frac{1}{2}\right)$
2. B.$\frac{\pi }{2}-{\cot }^{-1}\left(\frac{1}{2}\right)$
3. C.$\frac{\pi }{2}-{\tan }^{-1}\left(\frac{1}{2}\right)$
4. D.$\frac{\pi }{2}+{\tan }^{-1}\left(\frac{1}{2}\right)$

Consider two sets A and B, each containing three numbers in A.P. Let the sum and the product of the elements of A be 36 and p respectively and the sum and the product of the elements of B be 36 and $q$ respectively. Let d and D be the common differences of $A{P}^{\prime }s$ in $A$ and $B$ respectively such that $D=d+3,d>0$. If $\frac{p+q}{p-q}=\frac{19}{5}$, then $\mathrm{p}-\mathrm{q}$ is equal to

1. A.540
2. B.450
3. C.600
4. D.630

Let $a_1,a_2,a_3,\dots$ be in an A.P. such that ${\sum }_{k=1}^{12}{a}_{2k-1}=-\frac{72}{5}{a}_1,a_1\ne 0$. If ${\sum }_{k=1}^n{a}_k=0$, then $n$ is :

1. A.18
2. B.17
3. C.11
4. D.10

The number of terms of an A.P. is even; the sum of all the odd terms is 24 , the sum of all the even terms is 30 and the last term exceeds the first by $\frac{21}{2}$. Then the number of terms which are integers in the A.P. is :

1. A.6
2. B.4
3. C.8
4. D.10

If the sum of the first 10 terms of the series $\frac{4\cdot 1}{1+4\cdot 1^4}+\frac{4\cdot 2}{1+4\cdot 2^4}+\frac{4\cdot 3}{1+4\cdot 3^4}+\dots$ is $\frac{\mathrm{m}}{\mathrm{n}}$, where $\gcd (\mathrm{m},\mathrm{n})=1$, then $\mathrm{m}+\mathrm{n}$ is equal to ____.

Let $a_1,a_2,\dots ,a_{2024}$ be an Arithmetic Progression such that $a_1+(a_5+a_{10}+a_{15}+\dots +a_{2020})+a_{2024}=2233$. Then $a_1+a_2+a_3+\dots +a_{2024}$ is equal to ____.

Let ${\mathrm{T}}_{\mathrm{r}}$ be the ${\mathrm{r}}^{\mathrm{th}}$ term of an A.P. If for some $\mathrm{m},{\mathrm{T}}_{\mathrm{m}}=\frac{1}{25}$, ${\mathrm{T}}_{25}=\frac{1}{20}$, and $20{\sum }_{\mathrm{r}=1}^{25}{\mathrm{T}}_{\mathrm{r}}=13$, then $5\mathrm{m}{\sum }_{\mathrm{r}=\mathrm{m}}^{2\mathrm{m}}{\mathrm{T}}_{\mathrm{r}}$ is equal to

1. A.98
2. B.126
3. C.112
4. D.142

Let $S_n=\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\dots$ upto $n$ terms. If the sum of the first six terms of an A.P. with first term -p and common difference p is $\sqrt{2026S_{2025}}$, then the absolute difference between $20^{\text{th }}$ and $15^{\text{th }}$ terms of the A.P. is

1. A.$20$
2. B.$45$
3. C.$90$
4. D.$25$