Sequences, sigma notation, recurrence relations and modelling

Find the value of

(a) $\displaystyle\sum_{r=1}^{40}(3r-2)$ [2]

(b) $\displaystyle\sum_{r=5}^{12}2^{r}$ [3]

A sequence is defined by $$u_{1}=2,\qquad u_{n+1}=ku_{n}+3,\quad n\geqslant1$$ where $k$ is a constant.

(a) Find, in terms of $k$, an expression for $u_{3}$. [2]

Given that $u_{3}=5$,

(b) find the possible values of $k$. [2]

For the positive value of $k$ found in (b), the sequence converges to a limit $L$.

(c) Find the value of $L$. [2]

A company's annual profit is modelled as increasing by 4% each year. In Year 1 its profit is £250 000.

(a) Show that the model predicts a profit of £270 400 in Year 3. [1]

(b) Find the first year in which the model predicts the annual profit will exceed £400 000. [3]

(c) Find the total profit predicted by the model for Years 1 to 15 inclusive, giving your answer to the nearest £1000. [2]

(d) Give a reason why the model may not be suitable for predicting the profit in Year 40. [1]