To determine which statements or equations are correct about this geometric sequence, let's go through the details step by step.
### Step 1: Determining the Common Ratio
Given:
- [tex]\( a_6 = 0.00096 \)[/tex]
- [tex]\( a_7 = 0.000192 \)[/tex]
The common ratio [tex]\( r \)[/tex] of a geometric sequence can be calculated as:
[tex]\[ r = \frac{a_{7}}{a_{6}} \][/tex]
Substitute the given values:
[tex]\[ r = \frac{0.000192}{0.00096} \][/tex]
[tex]\[ r = 0.2 \][/tex]
So, statement A, [tex]\( r = 0.2 \)[/tex], is correct.
### Step 2: Determining the First Term ([tex]\( a_1 \)[/tex])
We know the general form of the [tex]\( n \)[/tex]-th term of a geometric sequence:
[tex]\[ a_n = a_1 \cdot r^{n-1} \][/tex]
Using [tex]\( a_6 \)[/tex], we can express it as:
[tex]\[ 0.00096 = a_1 \cdot (0.2)^{5} \][/tex]
Since [tex]\( 0.2^{5} = 0.00032 \)[/tex]:
[tex]\[ 0.00096 = a_1 \cdot 0.00032 \][/tex]
Solving for [tex]\( a_1 \)[/tex]:
[tex]\[ a_1 = \frac{0.00096}{0.00032} \][/tex]
[tex]\[ a_1 = 3 \][/tex]
So, statement C, [tex]\( a_1 = 3 \)[/tex], is correct.
### Step 3: Calculating the Sum of the First 20 Terms ([tex]\( S_{20} \)[/tex])
The sum of the first [tex]\( n \)[/tex] terms of a geometric sequence is given by:
[tex]\[ S_n = a_1 \cdot \frac{1 - r^n}{1 - r} \][/tex]
For [tex]\( n = 20 \)[/tex]:
[tex]\[ S_{20} = 3 \cdot \frac{1 - (0.2)^{20}}{1 - 0.2} \][/tex]
Since [tex]\( (0.2)^{20} \)[/tex] is a very small number, effectively close to 0:
[tex]\[ S_{20} \approx 3 \cdot \frac{1}{0.8} \][/tex]
[tex]\[ S_{20} \approx 3 \cdot 1.25 \][/tex]
[tex]\[ S_{20} \approx 3.75 \][/tex]
So, statement B, the sum of the first 20 terms rounded to the nearest hundredth is 3.75, is correct.
### Step 4: Calculating the Sum of the First 10 Terms ([tex]\( S_{10} \)[/tex])
Similarly, for [tex]\( n = 10 \)[/tex]:
[tex]\[ S_{10} = 3 \cdot \frac{1 - (0.2)^{10}}{1 - 0.2} \][/tex]
[tex]\[ (0.2)^{10} \][/tex] is a small number but not negligible, yielding a more exact value than the approximated result above.
Given the provided response ([tex]\(S_{10} \approx 3.75\)[/tex]), we see that:
So, statement F, the sum of the first 10 terms rounded to the nearest hundredth is 3.75, is correct.
### Final Verification:
Let's go through the statements:
- A: [tex]\( r = 0.2 \)[/tex] — Correct.
- B: The sum of the first 20 terms, rounded to the nearest hundredth, is 3.75 — Correct.
- C: [tex]\( a_1 = 3 \)[/tex] — Correct.
- D: [tex]\( a_1 = 15 \)[/tex] — Incorrect.
- E: [tex]\( r = 5 \)[/tex] — Incorrect.
- F: The sum of the first 10 terms, rounded to the nearest hundredth, is 18.75 — Incorrect (Sum is actually 3.75).
Thus, the correct statements are A, B, and C.
