Summary

This chapter built the algebra of sequences and series around two fundamental progressions and the ways they combine.

• A sequence is an ordered list $a_1,a_2,\ldots$ generated by a rule (explicit or recursive); a series is the sum of its terms, $S_n=\sum_{k=1}^n a_k$, with $a_n=S_n-S_{n-1}$ recovering any term from consecutive sums.
• An Arithmetic Progression has constant common difference $d$: general term $a_n=a+(n-1)d$; sum to $n$ terms $S_n=\dfrac n2[2a+(n-1)d]=\dfrac n2(a+l)$, derived by reversing the sum and adding it to itself.
• The arithmetic mean of $a,b$ is $A=\dfrac{a+b}{2}$; $n$ arithmetic means inserted between $a$ and $b$ use common difference $d=\dfrac{b-a}{n+1}$ and always sum to $nA$.
• A Geometric Progression has constant common ratio $r$: general term $a_n=ar^{n-1}$; sum to $n$ terms $S_n=\dfrac{a(1-r^n)}{1-r}$ for $r\ne1$ (and $S_n=na$ if $r=1$), derived by multiplying the sum by $r$ and subtracting.
• The geometric mean of positive $a,b$ is $G=\sqrt{ab}$; $n$ geometric means inserted between $a$ and $b$ use common ratio $r=(b/a)^{1/(n+1)}$ and always multiply to $G^n$.
• For positive $a,b$: $A\ge G$ always, with equality iff $a=b$ — proved from $(\sqrt a-\sqrt b)^2\ge0$; given $A$ and $G$, the numbers $a,b$ are recovered as the roots of $x^2-2Ax+G^2=0$.
• An Arithmetic-Geometric Progression multiplies an A.P. term-by-term by a G.P.; its sum to $n$ terms, derived by the same multiply-and-subtract idea (applied to a series that no longer fully cancels), is $S_n=\dfrac{a}{1-r}+\dfrac{dr(1-r^{n-1})}{(1-r)^2}-\dfrac{[a+(n-1)d]r^n}{1-r}$, with infinite-sum limit $\dfrac{a}{1-r}+\dfrac{dr}{(1-r)^2}$ when $|r|<1$. …