Practice section 13 of 17

The diagram shows $y=\ln(x+2)-x/5$. Its positive root is $\alpha$. Find $\alpha$. Positive terms satisfy $x_{n+1}=5\ln(x_n+2)$.

(i) Prove algebraically that convergence implies limit $\alpha$.

(ii) Use a calculator to describe the behaviour for $x_1=1$ and $x_1=20$.

(iii) By considering $x_{n+1}-x_n$, prove increase when $0<x_n<\alpha$ and decrease when $x_n>\alpha$.

Instruments A and B have bars of decreasing lengths.

(i) A has first bar $20$ cm and a GP of lengths; its 25th bar is $5$ cm. Show the total length is less than $357$ cm, however many bars there are.

B has the first 25 identical bars from A. (ii) Find its total $L$ and its 13th bar length.

(iii) A mistaken B has 25 AP lengths, the same total $L$, and last bar still $5$ cm. Find its common difference $d$ and longest bar.

A student with a knee problem plans swimming and jogging to prepare for a 2.4 km NAPFA run. On day 1 he jogs $200$ m and swims $200$ m. From day 2, jogging rises by $20\%$ of the previous day’s distance but is capped at $400$ m per day; swimming increases by $50$ m each day. Find the combined distance after 20 days, and the first day his combined daily distance reaches $2.4$ km.

A student deposits $10 on 1 January 2009 into an account paying $2\%$ compound interest on the last day of each month, then another $10 on the first day of each later month.

(a) How much interest has the original $10 earned after two years?

(b) What is the total balance after two years?

(c) After how many complete months does the total first exceed $2000?

Evaluate $\displaystyle\sum_{r=2}^{n}(2^{-r}+2nr+n^2)$, giving your answer in terms of $n$. [4]

Given $\sum_{r=1}^n(3r^2+3r+1)=(n+1)^3-1$, show $\sum_{r=1}^nr^2=\tfrac16n(n+1)(2n+1)$. [4]