Practice section 11 of 17

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.

The series $1+e^{-x}+e^{-2x}+\cdots$ has a sum to infinity. Show that $x>0$. When $x=10$, write $S_n$ and $S$, the finite and infinite sums. Show that $S-S_n<S/10^{100}$ requires $n>10\ln10$.

The seventh, third and first AP terms (nonzero common difference) are respectively the first three GP terms. Show that the geometric series converges. The seventh AP term is $3$. Find the smallest $n$ for which the first $n$ AP terms exceed the first $n$ GP terms by at least $100$.