Sum of n Terms

Q. A sequence $\left\{a_n\right\}$ is defined as
$a_n=\{\begin{array}{c}\frac{1}{n},\text{when}n\text{is odd}\\ -\frac{1}{n},\text{when}n\text{is even}\end{array}$
where $n\in N$. Then the sum of first four terms is

1. $\frac{7}{12}$
2. $\frac{5}{12}$
3. $0$
4. $\frac{3}{4}$

Q.

If ${}^n{P}_4=24·{}^n{C}_5$then the value of $n$ is

1. $10$

2. $15$

3. $9$

4. $5$

Q. The sum of the series 7+77+777+........of n terms is .

1. $\frac{7}{81}\left({10}^{n+1}-10-9n\right]$
2. $\frac{7}{81}\left({10}^{n+1}+10-9n\right]$
3. $\frac{7}{81}\left({10}^n-10-9n\right]$
4. $\frac{7}{81}\left({10}^n+10-9n\right]$

Q. If $S_n$ denotes the sum of first $n$ terms of an A.P. and $\frac{S_{3n}-S_{n-1}}{S_{2n}-S_{2n-1}}=31,$ then the value of $n$ is

Q. If $2f\left(xy\right)={\left(f\right(x\left)\right)}^y+{\left(f\right(y\left)\right)}^x\forall x,y\in R$ and $f\left(1\right)=3,$ then the value of $\sum _{r=1}^{10}f\left(r\right)$ is equal to

1. $\frac{3}{2}(3^{10}-1)$
2. $\frac{3}{2}(3^9-1)$
3. $\frac{1}{2}(3^{10}-1)$
4. $\frac{1}{2}(3^9-1)$

Q. There are $2n$ terms in an A.P., whose first term is $a$ and common difference is $d$. The sum of the odd terms is $24$ and the sum of the even terms is $30$. If the last term exceeds the first term by $10\frac{1}{2}$, then which of the following is (are) CORRECT?

1. $a=\frac{3}{2}$
2. $d=\frac{5}{2}$
3. $n=4$
4. $n=8$

Q.

If $2{\times }^n{C}_5=9\times {}^{n-2}C_5$, then the value of $n$ will be?

1. $7$

2. $10$

3. $9$

4. $5$

Q. A spiral is made up of successive semicircles, with centres alternately at $A$ and $B,$ starting with centre at $A$. If the radii are in sequence, $0.5\text{cm},1.0\text{cm},1.5\text{cm},2.0\text{cm},\dots$ as shown in figure below.


Then the total length of the spiral made by thirteen consecutive semicircles is
$(\text{Take}\pi =\frac{22}{7})$

1. $133\text{cm}$
2. $140\text{cm}$
3. $143\text{cm}$
4. $150\text{cm}$

Q. Concentric circles of radius $1,2,3,\dots ,100$ units are drawn such that the interior of the smallest circle is coloured red and the angular regions are coloured alternatively green and red, so that no two adjacent regions are of the same colour. Then the area of green region (in sq. units) is

1. $5000\pi$
2. $5050\pi$
3. $4950\pi$
4. $4900\pi$

Q. Let $A.P.(a,d)$ denotes the set of all the terms of an infinite arithmetic progression with first term $a$ and common difference $d>0$ . If $A.P.(1,4)\cap A.P.(1,5)\cap A.P.(3,3)=A.P.(a,d)$ then

1. $a=16$
2. $d=60$
3. sum of first $4$ terms of $A.P.(a,d)$ is $402$
4. $a+d=81$

Q. If the sum and product of $20$ terms in G.P. are $16$ and $1024$ respectively, then the sum of its reciprocals is

Q. The sum of an A.P series is $k\left(\frac{a+c}{c-b}\right),$ whose last two terms are $b$ and $c$ respectively. If $a$ be the first term of the series, then the value of $k$ is

1. $\frac{2c+b-a}{2}$
2. $\frac{2c+b+a}{2}$
3. $\frac{2c-b-a}{2}$
4. $\frac{2c-b+a}{2}$

Q.

Find the sum of the first 15 terms of the series 3 + 5 + 7 + 9 +. . . . . . n terms.___

Q. If the sum of $n$ terms of an A.P. is given by $S_n=a+bn+c{n}^2$, where $a,b,c$ are independent of $n$, then which of the following is (are) CORRECT?

1. $a=0$
2. Common difference of the A.P. must be $2b$
3. Common difference of the A.P. must be $2c$
4. First term of the A.P. is $b+c$

Q. The first term of an AP is $5$, last term is $45$ and the sum is $400,$ then the number of terms and common difference of the series are respectively :

1. $12,\frac{8}{3}$
2. $16,\frac{8}{3}$
3. $16,\frac{6}{11}$
4. $8,\frac{6}{11}$

Q. If $S_{2n}=3S_n$ and $S_{5n}=k{S}_{3n}$, where $S_n$ is the sum of $n$ terms of an A.P., then the value of $k$ is

1. $2$
2. $2.5$
3. $3$
4. $3.5$

Q. If $S_n$ denotes the sum of first $n$ terms of a non-constant A.P. whose first term is $a$ and $\frac{S_{nx}}{S_x}$ is independent of $x,$ then $S_p$ is equal to

1. $p^3$
2. $p^{2}a$
3. $p{a}^2$
4. $a^3$

Q. Let $S={1008}^2+{1007}^2+{1006}^2-{1005}^2-{1004}^2-{1003}^2+\cdots 6^2+5^2+4^2-3^2-2^2-1^2$, then

[Note: $1009$ is a prime number]

1. $S$ is divisible by $67$
2. $S$ is divisible by $5$
3. The number of factors of $S$ is $24$
4. The number of factors of $S$ is $16$

Q. In a school, students thought of planting trees in and around the school to reduce air pollution. It was decided that the number of trees that each section of each class will plant, will be same as the class, in which they are studying, e.g., a section of class $\text{I}$ will plant $1$ tree, a section of class $\text{II}$ will plant $2$ trees and so on till class $\text{XII}$, If there are three sections of each class, then the number of trees planted is

1. $78$
2. $234$
3. $39$
4. $156$

Q. The sum of first $20$ terms of the sequence $0.7,0.77,0.777,...,$ is

1. $\frac{7}{81}[179-{10}^{-20}]$
2. $\frac{7}{9}[99-{10}^{-20}]$
3. $\frac{7}{81}[179+{10}^{-20}]$
4. $\frac{7}{9}[99+{10}^{-20}]$

Q. Let $a_1,a_2,a_3,\dots$ be a G.P. with $a_1=a$ and common ratio $r,$ where $a$ and $r$ positive integers, then the number of ordered pairs $(a,r)$ such that $\sum _{r=1}^{12}{\log }_8{a}_r=2010$ is
(correct answer + 1, wrong answer - 0.25)

1. $40$
2. $42$
3. $44$
4. $46$

Q. If $\frac{2a+b+c}{a},\frac{a+2b+c}{b}$ and $\frac{a+b+2c}{c}$ are in A.P., then

1. $a,b,c$ are in A.P.
2. $\frac{1}{a},\frac{1}{b},\frac{1}{c}$ are in A.P.
3. $a,c,b$ are in A.P.
4. $\frac{1}{a},\frac{1}{c},\frac{1}{b}$ are in A.P.

Q. If ratio of the sum of $n$ terms of $2$ different A.P. is $\frac{3n+2}{4n+23}$, then the ratio of ${10}^{\text{th}}$ term is

1. $\frac{59}{99}$
2. $\frac{50}{99}$
3. $\frac{69}{99}$
4. $\frac{59}{79}$

Q. If $2+5+8+11+\cdots \text{upto}n\text{terms}=610$, then the value of $n$ is

1. $19$
2. $22$
3. $21$
4. $20$

Q. The value of $y=(0.36{)}^{{\log }_{0.25}(\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+\cdots \infty )}$ is

1. $0.9$
2. $0.8$
3. $0.6$
4. $0.25$

Q. If the sum of three consecutive of an A.P. is $-3$ and their product is $8$, then which of the following is/are correct?

1. The commmon difference can be $3$
2. One of the term is $-1$
3. Sum of the squares of the number is $21$.
4. Sum of the squares of the number is $25$.

Q. Let $m$ arithmetic means are inserted between $1$ and $31$. If the ratio of the $7^{\text{th}}$ mean and $(m-1{)}^{\text{th}}$ mean is $5:9$, then which of the following is/are correct?

1. The value of $m$ is $14$.
2. The value of $m$ is $16$.
3. The sum of all mean's is $256$.
4. The sum of all mean's is $224$.

Q. One side of an equilateral triangle is $24$ cm. The midpoints of its sides are joined to form another triangle whose midpoints are in turn joined to form still another triangle this process continues indefinitely. The sum of the perimeters of all the triangles is

1. $144\text{cm}$
2. $142\text{cm}$
3. $140\text{cm}$
4. $130\text{cm}$

Q. If the ratio of sum of first $n$ terms of two A.P.s is $2n+8:5n-3,$ then the ratio of $n^{\text{th}}$ terms of those two A.P.s is

1. $2n+5:5n+1$
2. $2n+3:5n-4$
3. $2n+18:5n-1$
4. $2n+18:5n+1$

Q. Concentric circles of radius $1,2,3,\dots ,100$ units are drawn such that the interior of the smallest circle is coloured red and the angular regions are coloured alternatively green and red, so that no two adjacent regions are of the same colour. Then the area of green region (in sq. units) is

1. $5000\pi$
2. $5050\pi$
3. $4950\pi$
4. $4900\pi$