Arun Sharma Solutions for Exercise 1: CAT

For how many integers $n$, will the inequality $\left(n-5\right)\left(n-10\right)-3\left(n-2\right)\le 0$ be satisfied?

If the square of the $7\text{th}$ term of an arithmetic progression with positive common difference equals the product of the $3\text{rd and} 17\text{th}$ terms, then the ratio of the first term to the common difference is 

Let $a_1,a_2,...,a_{3n}$ be an arithmetic progression with $a_1=3 \text{and} a_2=7$. If $a_1+a_2+...+a_{3n}=1830$, then what is the smallest positive integer $\text{'}m\text{'}$ such that $m\times \left(a_1+a_2+...+a_n\right)>1830$?

If the product of the three consecutive positive integers is $15600$ then the sum of the squares of these integers is 

Let $a_1,a_2,a_3,a_4,a_5$ be a sequence of five consecutive odd numbers. Consider a new sequence of five consecutive even numbers ending with $2a_3$. If the sum of the numbers in the new sequence is $450$, then $a_5$ is 

How many different pairs $\left(a,b\right)$ of positive integers are there such that $a\le b \text{and} \frac{1}{a}+\frac{1}{b}=\frac{1}{9}$?

An infinite geometric progression $a_1,a_2,a_3...$ has the property that $a_n=3\left(a_{n+1}+a_{n+2}+...\right)$ for every $n\ge 1$. If the sum $a_1+a_2+a_3+...=32$, then $a_5$ is

If $a_1=\frac{1}{2\times 5}, a_2=\frac{1}{5\times 8}, a_3=\frac{1}{8\times 11},...$ then $a_1+a_2+a_3+...+a_{100}$ is