Practice section 12 of 17

Find (i) $\sum_{k=0}^n(k+n+1)$ in terms of $n$; (ii) $\sum_{r=1}^N(4r+2^{-r})$ in terms of $N$; (iii) $\sum_{k=1}^{30}(a^{2k}+ka)$ in terms of $a$.

Given $\sum_{r=1}^nr(r+2)=n(n+1)(2n+7)/6$, find $\sum_{r=n}^{2n}r(r+2)$.

Given $\sum_{r=1}^n\dfrac{7r+10}{r(r+1)(r+2)}=6-\left(\dfrac5{n+1}+\dfrac2{n+2}\right)$, deduce the exact value of $\sum_{r=22}^\infty\dfrac{21r+30}{r(r+1)(r+2)}$.

Given $\sum_{r=2}^n\dfrac{3r^2+1}{(r^2-1)^3}=\dfrac12\left[\dfrac98-\dfrac1{n^3}-\dfrac1{(n+1)^3}\right]$, find $\sum_{r=3}^n\dfrac{3(r-1)^2+1}{[(r-1)^2-1]^3}$ in terms of $n$. Hence find its infinite sum.

Celine takes $500$ mg of Vitamin C once daily. The model assumes $20\%$ remains immediately before the next day’s pill. Let $x_n$ be the amount immediately after dose $n$, with $x_1$ immediately after the first dose.

(i) Write a recurrence. (ii) Show $x_n=(0.2)^{n-1}(x_1-625)+625$. (iii) Deduce the limit.

Positive real terms satisfy $x_{n+1}=2\sqrt{x_n+1}$.

(i) If the sequence converges, show its limit is $2+2\sqrt2$.

(ii) Prove $x_{n+1}<x_n$ whenever $x_n>2+2\sqrt2$.