Practice section 10 of 17

A zoology student models a jaguar chasing a white-tailed deer initially $d$ metres ahead. Both start together. The jaguar leaps $6$ m first, each later leap $0.1$ m shorter. The deer leaps $9$ m first, each later leap $5\%$ shorter.

(i) Find the deer’s total after $n$ leaps and deduce its maximum distance.

(ii) Assume both complete the same number of leaps in the same duration. For $d=11$ m, find the least $n$ for the jaguar to catch the deer within $n$ leaps.

A student drops a ball made of a new compound from $200$ cm onto a hard surface. Each rebound reaches $\frac89$ of the previous height.

(i) Find the distance travelled when it reaches the highest point after the fourth bounce, to the nearest centimetre.

(ii) The ball is considered to have stopped when a bounce first reaches a height below $0.01$ cm. Find the number of bounces and the total distance travelled, to the nearest centimetre.

A man deposits $200 at the beginning of each year into an account paying $5\%$ interest compounded annually. How much is in the account immediately after the tenth deposit?

A patient receives $10$ units of a drug by injection every $24$ hours. The fraction remaining in the body after $24$ hours is $e^{-1/5}$. If injections continue indefinitely, find the limiting number of units in the body just prior to an injection.

A sequence is generated by the recurrence relation $u_{n+1}=2u_n+1$ and the initial term $u_1=1$. Find the first five terms of this sequence and express $u_n$ in terms of $n$.

A sequence of real numbers satisfies $x_{n+1}=-e^{x_n}$ for $n\ge1$. Use a calculator to determine its behaviour when $x_1=-2$.