Arithmetic Progression

At a glance 10 formulas

1. 1nth term$T_n = a + (n - 1)\,d$$a = \text{first term},\ d = \text{common difference}$
2. 2Number of terms$n = \dfrac{l - a}{d} + 1$$l = \text{last term; the } {+}1 \text{ counts the first term}$
3. 3Sum of an AP$S_n = \dfrac{n}{2}\,(a + l)$$S_n = \dfrac{n}{2}\,[2a + (n - 1)d]$
4. 4Three in a row$b = \dfrac{a + c}{2}$$a, b, c \text{ consecutive; in a GP: } b^2 = ac$
5. 5Term from the sum$T_n = S_n - S_{n-1}$$S_n = pn^2 + qn \Rightarrow T_1 = p + q,\ d = 2p$
6. 6Odd count: middle term$S_{2k-1} = (2k - 1)\,T_k$$\text{e.g. } S_5 = 5\,T_3,\ \ S_9 = 9\,T_5$
7. 7Pairs from the ends$T_k + T_{n+1-k} = a + l$$k\text{th from the start} + k\text{th from the end}$
8. 8Equal sums cancel$S_m = S_n \Rightarrow S_{m+n} = 0$$m \ne n \text{ counts of terms from the start}$
9. 9Blocks of k terms$S_{2k} - 2S_k = k^2 d$$\text{each next block of } k \text{ terms is } k^2 d \text{ more}$
10. 10Common termsThe shared terms form a new AP: start at the first number in both lists and step by the LCM of the two common differences, up to the smaller last term.