Common Difference

Q. Where do we use BODMAS and DMAS rules precisely?

Q.

An AP starts with a positive fraction and every alternate term is an integer. If the sum of the first $11$terms is$33$, then the fourth term is what?

Q.

The first term of an A.P. is 2 and the last term is 50. The sum of all these terms is 442. Find the common difference.

Q.

if f(x)=cos [π]x+cos[πx] where [y] is the greatest integer function of y then f(π/2)=? Explain with steps please

Q.

Write the common difference of an A.P. whose nth term is xn + y.

Q.

If A and B are the sunsets of the universal set U, then show that

1. A is a subset of A union B

Q. If $2x,x+10,3x+2$ are in AP, find the value of x

Q. Sum of first 10 terms is-150 and the sum of next ten terms is -550. Find the AP

Q. If $\underset{n\to \infty }{lim}\sum _{r=1}^n\frac{a(1+3+5+\dots +(2r-1\left)\right)+(a-2)rn}{n\sum _{r=1}^n(2r-1)}=4$, then the value of $a$ is

1. $3$
2. $4$
3. $6$
4. $8$

Q.

Let the sequence $a_1,a_2,a_3.....a_n$ form an A.P. Then
$a_1^2-a_2^2+a_3^2-a_4^2+.....+a_{2n-1}^2-a_{2n}^2$ is equal to

1. None of these

2. $\frac{n}{2n-1}[a_1^2-a_{2n}^2]$

3. $\frac{n}{n-1}[a_0^2-a_{2n}^2]$

4. $\frac{n}{n+1}[a_1^2+a_{2n}^2]$

Q.

If $a_n,a_2,a_3,\dots \dots$ are in A.P., with common difference d, then the sum of the series $sind[sec{a}_{1}sec{a}_2+sec{a}_{2}sec{a}_3+\dots \dots +sec{a}_{n-1}sec{a}_n]$, is

1. $sec{a}_1-sec{a}_n$

2. $cosec{a}_1-cosec{a}_n$

3. $cot{a}_1-cot{a}_n$

4. $tan{a}_n-tan{a}_1$

Q. Find three numbers in A.P. whose sum is 15 and whose product is 105 ?

Q. Which term of A.P. $20,19\frac{1}{4},18\frac{1}{2}$, ..... is first negative term?

1. $!8$th
2. $27$th
3. $15$th
4. $28$th

Q. The number of terms common to the two A.P.s $2+5+8+11+.......+98$ and $3+8+13+18+....+198$ are

1. $33$
2. None of these
3. $40$
4. $7$

Q. If $a,(a-2)$ and $3a$ are in AP, then the value of $a$ is

1. $-3$
2. $-2$
3. $3$
4. $2$

Q. If the sum of n terms of an A.P. is $nA+n^{2}B,$ where A, B are constants, then its common difference will be
[MNR 1977]

1. A - B
2. A + B
3. 2A
4. 2B

Q. If $a_1,a_2,a_3$ .... a_{n} are in A. P., then the common difference is--

1. $a_2-a_1$
2. $a_4-a_3$
3. $a_n-a_{n-1}$
4. $a_7-a_6$

Q.

I + f - f' is an even integer, where I is an integer, f and f' are proper positive

fractions (0 < f, f' < 1). Then the integer I is an odd number.

1. False

2. True

Q. If |x|, |x-1|, |x+1| are the first three terms of an AP and d is the common difference between them, T(r) is the rth term then, A)T(5)= 9/4 B) d=0.5 C) d=0.25 D) T(10)= 19/4

Q. If the sum of n terms of an A.P. is $nA+n^{2}B,$ where A, B are constants, then its common difference will be
[MNR 1977]

1. A - B
2. A + B
3. 2A
4. 2B

Q. If $a,a_1$, $a_2$, $a_3$, ..., $a_{2n-1},b$ are in AP, $a$, $b_1$, $b_2$, $b_3$, ..., $b_{2n-1}$, $b$ are in GP and $a$, $c_1$, $c_2$, $c_3$, ..., $c_{2n-1}$, $b$ are in HP, where $a,b$ are positive, then the equation $a_n{x}^2-b_{n}x+c_n=0$ has its roots

1. real and unequal
2. real and equal
3. imaginary
4. none of these

Q. Which of the following equations is NOT equivalent to $10y^2=(x+2)(x-2)$ ?

1. $30y^2=3x^2-12$
2. $20y^2=(2x-4)(x+2)$
3. $5y^2=x^2-2$
4. $y^2=\frac{x^2-4}{10}$
5. ${10y}^2+4=x^2$

Q.

The sum of 4th and 8th terms of an A.P. is 24 and the sum of the 6th and 10th terms is 34. Find the first term and the common difference of the A.P.

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

1. 42
2. 47
3. 57
4. 52

Q.

Let [∙] represents the greatest integer function and f(x)=[tan² x], then

1. 
   

2. f(0)=1
3. f(x) is continuos at x=0
4. f(x) is non-differentiable at x=0

Q. Find the sum of the following series $1-\frac{n^2}{1^2}+\frac{n^2(n^2-1^2)}{1^2{.2}^2}-\frac{n^2(n^2-1^2)(n^2-2^2)}{1^2{.2}^2{.3}^2}+....$ to $n+1$ terms.

Q.

Let the sequence $a_1,a_2,a_3.....a_n$ form an A.P. Then
$a_1^2-a_2^2+a_3^2-a_4^2+.....+a_{2n-1}^2-a_{2n}^2$ is equal to

1. $\frac{n}{2n-1}[a_1^2-a_{2n}^2]$

2. $\frac{n}{n-1}[a_0^2-a_{2n}^2]$

3. $\frac{n}{n+1}[a_1^2+a_{2n}^2]$

4. None of these

Q.

Given an AP whose terms are all positive integers. The sum of its first nine terms is greater than 200 and less than 220. If the second term in it is 12, find the 4th term.