Practice section 16 of 17

The cubic $x^3-6x+1=0$ has a root $\alpha$ in $(0,1)$.

(i) Find $\alpha$ to 5 decimal places. [1]

(ii) For $x_{n+1}=(x_n^3+1)/6$, $x_1=0$, use a calculator to determine the sequence’s behaviour. [2]

(iii) Prove algebraically that, if the sequence converges to $\alpha$, its limit is the root of the given cubic. [2]

A storage tank loses $10\%$ of its water each day, then receives $90$ litres at the end of the day. Let $x_n$ be the volume after n days and $x_0$ its initial volume.

(i) Give a recurrence. [1]

(ii) Show $x_n=0.9^n(x_0-900)+900$. [3]

(iii) Deduce the limiting volume. [1]

The second, fifth and tenth terms of an arithmetic progression with nonzero common difference are the first three terms of a geometric progression.

(i) Find the common ratio. [3]

(ii) The first terms of the AP and GP are $7$ and $9$. Find the least n for which the GP’s n-th term exceeds the AP’s n-th term by at least $1000$. [3]

(a) A sequence has partial sums $S_n=n^2+6n$.

(i) Find and simplify $u_n$. [3]

(ii) Show it is an AP. [2]

(iii) Find $u_1+u_3+\cdots+u_{49}$. [2]

(b) A GP has nonzero first term a and nonzero ratio r with $-1<r<1$. Its (n−1)-th term equals the sum after, excluding, its n-th term.

(i) Show $r^2+r-1=0$. [3]

(ii) Deduce r. [1]

A class has $16$ boys with AP heights and $10$ girls with GP heights. The shortest boy and girl are each $150.0$ cm. The tallest boy is $180.0$ cm, and the fourth-shortest girl is $157.5$ cm.

(i) Show the girls’ common ratio is $1.05^{1/3}$ and find the tallest girl’s height to 1 d.p. [2]

(ii) Find how many girls are taller than $164.0$ cm. [3]

(iii) Find the girls’ mean height to 1 d.p. [3]

(iv) Find the class mean to 1 d.p. [2]

(i) Swimmer A swims $50$ m on day one and increases each subsequent day’s distance by $10\%$ of the previous day’s distance for $30$ days.

(a) Show her day-30 distance is $793$ m to the nearest metre. [2]

(b) Find her total in 30 days to the nearest metre. [2]

(ii) Swimmer B starts at $50$ m and adds $5$ m each day. Find the least number of days for B’s total to exceed A’s total in (i)(b). [5]