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 \(\( r \)\).
- Given: \(\( a_1 = 1.1 \)\) meters (growth in the first year).
- We know the tree grows \(\( 0.62 \)\) meters in the third year, i.e., \(\( a_3 = 0.62 \)\) meters.
2. Express the terms in the sequence mathematically:
- For a geometric sequence, the \(\( n \)\)-th term \(\( a_n \)\) is given by:
\(\[
a_n = a_1 \cdot r^{(n-1)}
\]\)
3. Find the ratio \(\( r \)\):
- We have the first term \(\( a_1 = 1.1 \)\) meters.
- The third term \(\( a_3 = 0.62 \)\) meters.
- Using the formula for the \(\( n \)\)-th term, we can express \(\( a_3 \)\) as:
\(\[
a_3 = a_1 \cdot r^2
\]\)
- Substituting the known values:
\(\[
0.62 = 1.1 \cdot r^2
\]\)
4. Solve for \(\( r \)\):
- Divide both sides by \(\( 1.1 \)\):
\(\[
r^2 = \frac{0.62}{1.1}
\]\)
- Calculate the right-hand side:
\(\[
r^2 \approx 0.5636
\]\)
- Take the square root of both sides to find \(\( r \)\):
\(\[
r \approx \sqrt{0.5636} \approx 0.75
\]\)
5. Determine the second year's growth using \(\( r \)\):
- The second term \(\( a_2 \)\) in the sequence is:
\(\[
a_2 = a_1 \cdot r
\]\)
- Substituting the values for \(\( a_1 \)\) and \(\( r \)\):
\(\[
a_2 = 1.1 \cdot 0.75 \approx 0.825 \text{ meters}
\]\)
6. Represent the yearly growth sequence:
- First year’s growth: \(\( a_1 = 1.1 \)\) meters
- Second year’s growth: \(\( a_2 \approx 0.825 \)\) meters
- Third year’s growth: \(\( a_3 = 0.62 \)\) 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
