Why not start out by writing a linear relationship for the plant height?
h(m) = 10 + 3m, where h is the height in mm and m is the time in years, starting from zero (0).
After 8 months, the plant height is h(8) = 10 mm + 3(8) mm, or 34 mm.
After 3 years (36 months), the plant ht. is h(36) = 118 mm.
Plug in the numbers and solve:
8.5(<u>3</u>) + 7.9(<u>4</u>) = <u>25.5</u> + 31.6
= <u>57.1</u>
You're looking for a number <em>w</em> such that the numbers
{1 + <em>w</em>, 7 + <em>w</em>, 15 + <em>w</em>}
form a geometric sequence, which in turn means there is a constant <em>r</em> for which
7 + <em>w</em> = <em>r</em> (1 + <em>w</em>)
15 + <em>w</em> = <em>r</em> (7 + <em>w</em>)
Solving for <em>r</em>, we get
<em>r</em> = (7 + <em>w</em>) / (1 + <em>w</em>) = (15 + <em>w</em>) / (7 + <em>w</em>)
Solve this for <em>w</em> :
(7 + <em>w</em>)² = (15 + <em>w</em>) (1 + <em>w</em>)
49 + 14<em>w</em> + <em>w</em> ² = 15 + 16<em>w</em> + <em>w</em> ²
2<em>w</em> = 34
<em>w</em> = 17
Then the three terms in the sequence are
{18, 24, 32}
and indeed we have 24/18 = 4/3 and 32/24 = 4/3.
