The number of $5 -$tuples $(a, b, c, d, e)$ of positive integers such that

$I.$ $a, b, c, d, e$ are the measures of angles of a convex pentagon in degrees

$II$. $a \leq b \leq c \leq d \leq e$

$III.$ $a, b, c, d, e$ are in arithmetic progression is

Write the first five terms of the following sequence and obtain the corresponding series :

$a_{1}=a_{2}=2, a_{n}=a_{n-1}-1, n\,>\,2$

The sums of $n$ terms of two arithmetic progressions are in the ratio $5 n+4: 9 n+6 .$ Find the ratio of their $18^{\text {th }}$ terms.

Let $\frac{1}{{{x_1}}},\frac{1}{{{x_2}}},\frac{1}{{{x_3}}},.....,$  $({x_i} \ne \,0\,for\,\,i\, = 1,2,....,n)$  be in $A.P.$  such that  $x_1 = 4$ and $x_{21} = 20.$ If $n$  is the least positive integer for which $x_n > 50,$  then $\sum\limits_{i = 1}^n {\left( {\frac{1}{{{x_i}}}} \right)} $  is equal to.

Find the $9^{\text {th }}$ term in the following sequence whose $n^{\text {th }}$ term is $a_{n}=(-1)^{n-1} n^{3}$

The sum of all the elements in the set $\{\mathrm{n} \in\{1,2, \ldots \ldots ., 100\} \mid$ $H.C.F.$ of $n$ and $2040$ is $1\,\}$ is equal to $.....$