Sequences and Series

Let $a,b,c$ be positive integers in arithmetic progression such that the equation
$$ax^2+bx+c=0$$
has only integer solutions.

Then which of the following statements is (are) TRUE ?

Let $\alpha,\ \alpha+2$, $\alpha\in\mathbb Z$, be the roots of the quadratic equation
$$x(x+2)+(x+1)(x+3)+(x+2)(x+4)+\dots+(x+n-1)(x+n+1)=4n$$
for some $n\in\mathbb N$. Then $n+\alpha$ is equal to :

Let $A$ be the set of first 101 terms of an A.P., whose first term is 1 and the common difference is 5 and let $B$ 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 $A\cap B$, which are divisible by 3, is :

If $\vec a$ and $\vec b$ are two vectors such that $|\vec a|=2$ and $|\vec b|=3$, then the maximum value of $3\left|\left(3\vec a+2\vec b\right)\right|+4\left|\left(3\vec a-2\vec b\right)\right|$ is :

If $\displaystyle\sum_{k=1}^na_k=6n^3$, then $\displaystyle\sum_{k=1}^6\left(\frac{a_{k+1}-a_k}{36}\right)^2$ is equal to ________.

Let $a_1,a_2,a_3,\dots$ be an A.P. and $g_1=a_1,g_2,g_3,\dots$ be an increasing G.P. If $a_1=a_2+g_2=1$ and $a_3+g_3=4$, then $a_{10}+g_5$ is equal to :

The sum $\dfrac{1^3}1+\dfrac{1^3+2^3}{1+3}+\dfrac{1^3+2^3+3^3}{1+3+5}+\cdots$ up to 8 terms, is :

A man throws a fair coin repeatedly. He gets 10 points for each head he throws and 5 points for each tail he throws. If the probability that he gets exactly 30 points is $\dfrac mn$, $\gcd(m,n)=1$, then $m+n$ is equal to :

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

Let $\alpha=\dfrac14+\dfrac18+\dfrac1{16}+\dots\infty$ and $\beta=\dfrac13+\dfrac19+\dfrac1{27}+\dots\infty$. Then the value of $(0.2)^{\log_{\sqrt5}(\alpha)}+(0.04)^{\log_5(\beta)}$ is equal to:

Let for some $\alpha\in\mathbb R$, $f:\mathbb R\to\mathbb R$ be a function satisfying $f(x+y)=f(x)+2y^2+y+\alpha xy$ for all $x,y\in\mathbb R$. If $f(0)=-1$ and $f(1)=2$, then the value of $\displaystyle\sum_{n=1}^5\left(\alpha+f(n)\right)$ is:

From a month of 31 days, 3 different dates are selected at random. If the probability that these dates are in an increasing A.P. is equal to $\dfrac ab$, where $a,b\in\mathbb N$ and $\gcd(a,b)=1$, then $a+b$ 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:

$\displaystyle\sum_{n=1}^{10}\left(\frac{528}{n(n+1)(n+2)}\right)$ is equal to:

Let $\alpha,\beta$ be the roots of the equation $x^2-x+p=0$ and $\gamma,\delta$ be the roots the equation $x^2-4x+q=0$; $p,q\in\mathbb Z$. If $\alpha,\beta,\gamma,\delta$ are in G.P., then $|p+q|$ equals :

If $f:\mathbf N\to\mathbf Z$ is defined by
$$f(n)=\begin{vmatrix}n&-1&-5\\-2n^2&3(2k+1)&2k+1\\-3n^3&3k(2k+1)&3k(k+2)+1\end{vmatrix},\quad k\in\mathbf N,$$
and $\displaystyle\sum_{n=1}^{k}f(n)=98$, then k is equal to :

If the sum of the first 10 terms of the series $\dfrac1{1+1^4\times4}+\dfrac2{1+2^4\times4}+\dfrac3{1+3^4\times4}+\dfrac4{1+4^4\times4}+\dots$ is $\dfrac mn$, $\gcd(m,n)=1$, then $m+n$ is equal to :

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 :

Let $\left(2^{1-a}+2^{1+a}\right)$, $f(a)$, $\left(3^a+3^{-a}\right)$ be in A.P. and $\alpha$ be the minimum value of $f(a)$. Then the value of the integral $\displaystyle\int_{\log_e(\alpha-1)}^{\log_e(\alpha)}\frac{dx}{\left(e^{2x}-e^{-2x}\right)}$ is :

The value of $1^3-2^3+3^3-\dots+15^3$ is: