The sum of the first terms, ${\mathrm{S}}_{\mathrm{n}}$, of a particular arithmetic progression is given by ${\mathrm{S}}_{\mathrm{n}}=12\mathrm{n}-2{\mathrm{n}}^2$. Find the first term and the common difference.

An arithmetic sequence has first term $a$ and common difference $d$. The sum of first $20$ terms is $7$ times the sum of the first five terms. Find $d$ in terms of $a$.

An arithmetic sequence has first term a and common difference d. The sum of first 20 terms is 7 times the sum of the first five terms. Find the 65th term in terms of a.

The first term of an arithmetic progression is ${\sin }^2\mathrm{x}$, and the second term is $1$. Write down an expression, in terms of $\sin$, for the fifth term of this progression.

The first term of an arithmetic progression is ${\sin }^2\mathrm{x}$, and the second term is $1$. Show that the sum of the first ten terms of this progression is $10+345 {\cos }^2\mathrm{x}$. 

The sum of the digits in the number $67$ is $13 (as 6+7=3)$. Show that the sum of the digits of the integers from $19 \mathrm{to} 21$ is $15$.