examday

/TMUA

Consider the sequence

• $a_1=k$;
• $a_n=-\dfrac12a_{n-1}+3$.

for some number $k$.

Which of the following must be true?

• I. If there exists $n>1$ where $a_n<-2$, then $k>10$.
• II. If $k>1$, then for all $n\ge1$, $|a_n|\le|a_1|$.
• III. For all $n\ge1$, $|a_{n+1}-2|\le|a_n-2|$.