Sequences & Series · JEE Main previous year questions

If the sum and product of the first three terms in an A.P. are 33 and 1155, respectively, then a value of its $11^{th}$ term is :-

1. A.−25
2. B.−36
3. C.25
4. D.−35

The sum of the series 1 + 2 × 3 + 3 × 5 + 4 × 7 +.... upto 11th term is :-

1. A.945
2. B.916
3. C.915
4. D.946

If the lengths of the sides of a triangle are in A.P. and the greatest angle is double the smallest, then a ratio of lengths of the sides of this triangle is :

1. A.5 : 9 : 13
2. B.5 : 6 : 7
3. C.4 : 5 : 6
4. D.3 : 4 : 5

The product of three consecutive terms of a G.P. is 512. If 4 is added to each of the first and the second of these terms, the three terms now form an A.P. Then the sum of the original three terms of the given G.P. is :

1. A.36
2. B.28
3. C.32
4. D.24

If ${}^n{\mathrm{C}}_4$, ${}^n{\mathrm{C}}_5$ and ${}^n{\mathrm{C}}_6$ are in A.P., then n can be :

1. A.11
2. B.12
3. C.9
4. D.14

If the sum of the first 15 terms of the series $(\frac{3}{4}{)}^3+(1\frac{1}{2}{)}^3+(2\frac{1}{4}{)}^3+3^3+(3\frac{3}{4}{)}^3+\dots$ is equal to $225k$, then k is equal to :

1. A.9
2. B.108
3. C.27
4. D.54

If $19^{\text{th}}$ term of a non-zero A.P. is zero, then its $(49^{\text{th}}\text{ term}):(29^{\text{th}}\text{ term})$ is :

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

If a, b, c be three distinct real numbers in G.P. and a + b + c = xb , then x cannot be

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

Let $a_1,a_2,\dots ,a_{30}$ be an A.P., $$S=\sum _{i=1}^{30}{a}_i\text{and}T=\sum _{i=1}^{15}{a}_{(2i-1)}.$$ If $a_5=27$ and $S-2T=75$, then $a_{10}$ is equal to :

1. A.47
2. B.42
3. C.52
4. D.57

Let $\frac{1}{x_1},\frac{1}{x_2},\dots ,\frac{1}{x_n}$ $(x_i\ne 0\text{ for }i=1,2,\dots ,n)$ be in A.P. such that $x_1=4$ and $x_{21}=20$. If n is the least positive integer for which $x_n>50$, then ${\sum }_{i=1}^n\left(\frac{1}{x_i}\right)$ is equal to :

1. A.$\frac{1}{8}$
2. B.3
3. C.$\frac{13}{8}$
4. D.$\frac{13}{4}$

The sum of the first 20 terms of the series $1+\frac{3}{2}+\frac{7}{4}+\frac{15}{8}+\frac{31}{16}+\dots$, is :

1. A.$38+\frac{1}{2^{19}}$
2. B.$38+\frac{1}{2^{20}}$
3. C.$39+\frac{1}{2^{20}}$
4. D.$39+\frac{1}{2^{19}}$

If $x_1,x_2,\dots ,x_n$ and $\frac{1}{h_1},\frac{1}{h_2},\dots ,\frac{1}{h_n}$ are two A.P.s such that $x_3=h_2=8$ and $x_8=h_7=20$, then $x_5.h_{10}$ equals :

1. A.2560
2. B.2650
3. C.3200
4. D.1600

If b is the first term of an infinite G.P. whose sum is five, then b lies in the interval :

1. A.$(-\infty ,-10]$
2. B.$(-10,0)$
3. C.$(0,10)$
4. D.$[10,\infty )$

Let $a_1,a_2,a_3,\dots \dots \dots ,a_{49}$ be in A.P. such that ${\sum }_{k=0}^{12}{a}_{4k+1}=416$ and $a_9+a_{43}=66$. $a_1^2+a_2^2+\dots \dots \cdots +a_{17}^2=140m$, then m is equal to

1. A.$33$
2. B.$66$
3. C.$68$
4. D.$34$

Let A be the sum of the first 20 terms and B be the sum of the first 40 terms of the series $1^2+2\cdot 2^2+3^2+2\cdot 4^2+5^2+2\cdot 6^2\dots \dots \dots \dots$ If $\mathrm{B}-2A=100\lambda$, then $\lambda$ is equal to

1. A.$496$
2. B.$232$
3. C.$248$
4. D.$464$

If three positive numbers a, b and c are in A.P. such that $\mathrm{abc}=8$, then the minimum possible value of b is :

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

A tangent to the curve, $y=f(x)$ at $\mathrm{P}(x,y)$ meets x-axis at A and y-axis at B. If $\mathrm{AP}:\mathrm{BP}=1:3$ and $f(1)=1$, then the curve also passes through the point :

1. A.$(\frac{1}{3},24)$
2. B.$(\frac{1}{2},4)$
3. C.$(2,\frac{1}{8})$
4. D.$(3,\frac{1}{28})$

If the arithmetic mean of two numbers $a$ and $b$, $a>b>0$, is five times their geometric mean, then $\frac{a+b}{a-b}$ is equal to :

1. A.$\frac{\sqrt{6}}{2}$
2. B.$\frac{3\sqrt{2}}{4}$
3. C.$\frac{7\sqrt{3}}{12}$
4. D.$\frac{5\sqrt{6}}{12}$

If the sum of the first n terms of the series $\sqrt{3}+\sqrt{75}+\sqrt{243}+\sqrt{507}+\dots \dots$ is $435\sqrt{3}$, then n equals :

1. A.$18$
2. B.$15$
3. C.$13$
4. D.$29$

If the $2^{nd},5^{th}and{9}^{th}$ terms of a non-constant A.P. are in G.P., then the common ratio of this G.P. is :

1. A.$1$
2. B.$\frac{7}{4}$
3. C.$\frac{8}{5}$
4. D.$\frac{4}{3}$

If the sum of the first ten terms of the series ${\left(1\frac{3}{5}\right)}^2+{\left(2\frac{2}{5}\right)}^2+{\left(3\frac{1}{5}\right)}^2+4^2+{\left(4\frac{4}{5}\right)}^2+\dots \dots$ is $\frac{16}{5}m$, then m is equal to :

1. A.$100$
2. B.$99$
3. C.$102$
4. D.$101$

If $m$ is the A.M. of two distinct real numbers $l$ and $n$ ($l,n>1$) and $G_1,G_2$ and $G_3$ are three geometric means between $l$ and $n$, then $G_1^4+2G_2^4+G_3^4$ equals:

1. A.$4lm{n}^2$
2. B.$4l^2{m}^2{n}^2$
3. C.$4l^{2}mn$
4. D.$4l{m}^{2}n$

The sum of first 9 terms of the series. $$\frac{1^3}{1}+\frac{1^3+2^3}{1+3}+\frac{1^3+2^3+3^3}{1+3+5}+\dots \dots$$

1. A.$142$
2. B.$192$
3. C.$71$
4. D.$96$

Three positive numbers form an increasing G.P. If the middle term in this G.P. is doubled, the new numbers are in A.P. then the common ratio of the G.P. is :

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

If the A.M. between $p^{\mathrm{th}}$ and $q^{\mathrm{th}}$ terms of an A.P. is equal to the A.M. between $r^{\mathrm{th}}$ and $s^{\mathrm{th}}$ terms of the same A.P., then $p+q$ is equal to

1. A.$r+s-1$
2. B.$r+s-2$
3. C.$r+s+1$
4. D.$r+s$