Geometric progression

Let $a,b,c$ be positive integers such that $\dfrac ba$ is an integer. If $a,b,c$ are in geometric progression and the arithmetic mean of $a,b,c$ is $b+2$, then the value of
$$\frac{a^2+a-14}{a+1}$$
is

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 :

The coefficient of $x^{50}$ in the binomial expansion of
$$(1+x)^{1000}+x(1+x)^{999}+x^2(1+x)^{998}+\dots+x^{1000}$$
is :

In a geometric progression, if the ratio of the sum of first 5 terms to the sum of their reciprocals is 49, and the sum of the first and the third term is 35. Then the first term of this geometric progression is :

The least positive integer n such that $1-\dfrac23-\dfrac2{3^2}-\dots-\dfrac2{3^{n-1}}<\dfrac1{100}$, is :

Let $b=6$, with $a$ and $c$ satisfying (E). If $\alpha$ and $\beta$ are the roots of the quadratic equation $ax^2+bx+c=0$, then
$$\sum_{n=0}^\infty\left(\frac1\alpha+\frac1\beta\right)^n$$
is

Let $S_k$, $k=1,2,\ldots,100$, denote the sum of the infinite geometric series whose first term is $\dfrac{k-1}{k!}$ and the common ratio is $\dfrac1k$. Then the value of $\dfrac{100^2}{100!}+\displaystyle\sum_{k=1}^{100}\left|(k^2-3k+1)S_k\right|$ is

Suppose four distinct positive numbers $a_1,a_2,a_3,a_4$ are in G.P. Let $b_1=a_1$, $b_2=b_1+a_2$, $b_3=b_2+a_3$ and $b_4=b_3+a_4$.

STATEMENT-1: The numbers $b_1,b_2,b_3,b_4$ are neither in A.P. nor in G.P.

and

STATEMENT-2: The numbers $b_1,b_2,b_3,b_4$ are in H.P.