11 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.
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\includegraphics[alt={},max width=\textwidth]{faeaf2aa-ed4e-4926-b402-40c4c9aad479-5_501_1102_431_504}
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\caption{Fig. 11}
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- 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.