The general term and the sum of the first $n$ terms of an arithmetic progression, with problems on finding $a$, $d$ and $n$ and summing numbers in a range.
In an arithmetic progression (AP) each term is found by adding the same number to the one before. That number is the common difference $d$.
$$U_{r+1} - U_r = d \quad\text{for every } r .$$
For example $3, 7, 11, 15, \dots$ has first term $a = 3$ and $d = 4$; $20, 17, 14, \dots$ has $d = -3$ (a decreasing AP). To test whether a sequence is an AP, check that the differences between consecutive terms are all equal.
Start at $a$ and add $d$ each time: $U_1 = a$, $U_2 = a + d$, $U_3 = a + 2d$. To reach the $n$th term we add $d$ exactly $n - 1$ times: $$U_n = a + (n - 1)d .$$
Example: The third term of an AP is $11$ and the eighth term is $31$. Find the first term and the common difference. Then $a + 2d = 11$ and $a + 7d = 31$. Subtracting, $5d = 20$, so $d = 4$ and $a = 11 - 8 = 3$. Check: $U_8 = 3 + 7(4) = 31$.
Write $S_n$ forwards and backwards and add: $$\begin{aligned} S_n &= a + (a+d) + \dots + (l - d) + l,\ S_n &= l + (l-d) + \dots + (a+d) + a. \end{aligned}$$ Each of the $n$ vertical pairs adds to $a + l$, so $2S_n = n(a + l)$. Putting $l = a + (n-1)d$, $$S_n = \frac{n}{2}(a + l) = \frac{n}{2}\left[2a + (n-1)d\right].$$ Use the first form when you know the last term, the second when you know $d$.
Example: For the AP of the example above ($a = 3$, $d = 4$) find $S_{20}$. $S_{20} = \dfrac{20}{2}\left[2(3) + 19(4)\right] = 10 \times 82 = 820$.
Example: How many terms of $5 + 9 + 13 + \dots$ are needed for the sum to be $275$? Here $a = 5$ and $d = 4$, so $S_n = \dfrac{n}{2}\left[10 + 4(n-1)\right] = n(2n + 3)$. Set $n(2n + 3) = 275$, that is $2n^2 + 3n - 275 = 0$, which factorises as $(n - 11)(2n + 25) = 0$. Since $n$ is a positive integer, $n = 11$.
To sum all the multiples of $7$ between $100$ and $300$, find the first and last multiples ($105$ and $294$); they form an AP with $d = 7$, so the number of terms is $n = \dfrac{294 - 105}{7} + 1 = 28$ and the sum is $\dfrac{28}{2}(105 + 294) = 5586$.
Exam tip: Write down "$a = \dots$, $d = \dots$" first, then pick the formula. When a problem gives a sum and asks for the number of terms, you get a quadratic in $n$; reject any negative or non-integer root and say why. "Show that the terms are in AP" is answered by $U_{r+1} - U_r = $ constant, not by listing a few terms.