How many times does the digit $5$ appear when writing natural numbers from $1$ to $100$?

The number of terms common to the two arithmetic progressions $3, 7, 11, \ldots, 407$ and $2, 9, 16, \ldots, 709$ is

If $t_n = \frac{1}{4}(n + 2)(n + 3)$ for $n = 1, 2, 3, \dots$,then $\frac{1}{t_1} + \frac{1}{t_2} + \frac{1}{t_3} + \dots + \frac{1}{t_{2003}} = $

The sum of the series $\frac{3}{1! + 2! + 3!} + \frac{4}{2! + 3! + 4!} + \frac{5}{3! + 4! + 5!} + \dots + \frac{2008}{2006! + 2007! + 2008!}$ is equal to

The sum of the first two terms of a $G.P.$ is $1$ and every term of this series is twice its previous term. Then,the first term will be:

Given an $A.P.$ 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 $12$,then its $4^{th}$ term is