4 There is a flowerhead at the end of each stem of an oleander plant. The next year, each flowerhead is replaced by three stems and flowerheads, as shown in Fig. 11.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f291e6e3-975e-4d1e-aab6-67308f305da2-2_517_1116_356_455}
\captionsetup{labelformat=empty}
\caption{Fig. 11}
\end{figure}
- How many flowerheads are there in year 5 ?
- How many flowerheads are there in year \(n\) ?
- As shown in Fig. 11, the total number of stems in year 2 is 4, (that is, 1 old one and 3 new ones). Similarly, the total number of stems in year 3 is 13 , (that is, \(1 + 3 + 9\) ).
Show that the total number of stems in year \(n\) is given by \(\frac { 3 ^ { n } - 1 } { 2 }\).
- Kitty's oleander has a total of 364 stems. Find
(A) its age,
(B) how many flowerheads it has. - Abdul's oleander has over 900 flowerheads.
Show that its age, \(y\) years, satisfies the inequality \(y > \frac { \log _ { 10 } 900 } { \log _ { 10 } 3 } + 1\).
Find the smallest integer value of \(y\) for which this is true.