Medium Practice — Set 2

Quick Recap — GP Sums & Special Series

• GP sum: $S_n=\dfrac{a(r^n-1)}{r-1}$; infinite: $\dfrac{a}{1-r}$ ($|r|<1$).
• Inserting $k$ GMs between $a,b$: ratio $r=\big(\tfrac ba\big)^{1/(k+1)}$.
• $\sum n^2=\tfrac{n(n+1)(2n+1)}{6}$, $\sum n^3=\Big(\tfrac{n(n+1)}{2}\Big)^2$.
• AM $\ge$ GM for positive numbers; $a,b,c$ in GP $\iff b^2=ac$.