Certainly! Let’s determine the yearly growth of a tree represented as a geometric sequence based on the given data: the tree grows 1.1 meters in the first year and 0.62 meters in the third year.
### Step-by-Step Solution:
1. Understand the terms of a geometric sequence:
- A geometric sequence is one in which each term is obtained by multiplying the previous term by a constant ratio [tex]\( r \)[/tex].
- Given: [tex]\( a_1 = 1.1 \)[/tex] meters (growth in the first year).
- We know the tree grows [tex]\( 0.62 \)[/tex] meters in the third year, i.e., [tex]\( a_3 = 0.62 \)[/tex] meters.
2. Express the terms in the sequence mathematically:
- For a geometric sequence, the [tex]\( n \)[/tex]-th term [tex]\( a_n \)[/tex] is given by:
[tex]\[
a_n = a_1 \cdot r^{(n-1)}
\][/tex]
3. Find the ratio [tex]\( r \)[/tex]:
- We have the first term [tex]\( a_1 = 1.1 \)[/tex] meters.
- The third term [tex]\( a_3 = 0.62 \)[/tex] meters.
- Using the formula for the [tex]\( n \)[/tex]-th term, we can express [tex]\( a_3 \)[/tex] as:
[tex]\[
a_3 = a_1 \cdot r^2
\][/tex]
- Substituting the known values:
[tex]\[
0.62 = 1.1 \cdot r^2
\][/tex]
4. Solve for [tex]\( r \)[/tex]:
- Divide both sides by [tex]\( 1.1 \)[/tex]:
[tex]\[
r^2 = \frac{0.62}{1.1}
\][/tex]
- Calculate the right-hand side:
[tex]\[
r^2 \approx 0.5636
\][/tex]
- Take the square root of both sides to find [tex]\( r \)[/tex]:
[tex]\[
r \approx \sqrt{0.5636} \approx 0.75
\][/tex]
5. Determine the second year's growth using [tex]\( r \)[/tex]:
- The second term [tex]\( a_2 \)[/tex] in the sequence is:
[tex]\[
a_2 = a_1 \cdot r
\][/tex]
- Substituting the values for [tex]\( a_1 \)[/tex] and [tex]\( r \)[/tex]:
[tex]\[
a_2 = 1.1 \cdot 0.75 \approx 0.825 \text{ meters}
\][/tex]
6. Represent the yearly growth sequence:
- First year’s growth: [tex]\( a_1 = 1.1 \)[/tex] meters
- Second year’s growth: [tex]\( a_2 \approx 0.825 \)[/tex] meters
- Third year’s growth: [tex]\( a_3 = 0.62 \)[/tex] meters
Therefore, the yearly growth of the tree, represented as a geometric sequence with the indicated values and sensible rounding, can be described as follows:
- First year: 1.1 meters
- Second year: 0.83 meters (rounded to two decimal places for simplicity)
- Third year: 0.62 meters
