Find the number of terms and the sum of each of these arithmetic series.

$4+1+\left(-2\right)+... \left(38 \mathrm{terms}\right)$.

Find the sum of each of these arithmetic series.

$\frac{1}{3}+\frac{1}{2}+\frac{2}{3}+... \left(20 \mathrm{terms}\right)$.

Find the sum of each of these arithmetic series.

$-\mathrm{x}-5\mathrm{x}-9\mathrm{x}-... \left(40 \mathrm{terms}\right)$.

The first term of an arithmetic progression is $15$ and the sum of the first $20$ terms is $1630$. Find the common difference.

In an arithmetic progression, the first term is $-27$, the ${16}^{\mathrm{th}}$ term is $78$ and the last term is $169$. Find the common difference and the number of terms.
 

In an arithmetic progression, the first term is $-27$, the ${16}^{\mathrm{th}}$ term is $78$ and the last term is $169$. Find the sum of the terms in this progression.