Discussion questions · Section 1 of 3

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 $x_1=-2$.

Find the sum of all positive integers less than $70$ which do not contain the digit $3$.

Sonia buys a car with an interest-free loan of $P$ dollars. Monthly instalments start at $1200 and rise by $50 each month ($1250, $1300, …). The final instalment is $3250 and the loan is repaid in $k$ instalments. Find $k$ and $P$.

A string has 31 pearls, with the largest and most expensive in the middle. From one end each pearl is worth $100 more than its predecessor up to the middle; from the other end each is worth $150 more. The whole string costs $63,000. Find the middle pearl’s value.

The partial sum is $S_n=\dfrac1a[1-(a-1)^n]$, where $a\ne0,1$. Find the nth term $u_n$, and prove that $S_n$ is a geometric series.