Answer :
Certainly! To find the sum of the first nine terms of a geometric sequence where the 3rd term is 45 and the 6th term is 1215, let's use the properties and formulas of geometric sequences.
### Step-by-Step Solution:
1. Understand the Given Information:
- The 3rd term ([tex]\(a_3\)[/tex]) is 45.
- The 6th term ([tex]\(a_6\)[/tex]) is 1215.
2. Formulate the General Form of the nth Term:
The nth term of a geometric sequence can be expressed as:
[tex]\[ a_n = a \cdot r^{n-1} \][/tex]
where:
- [tex]\(a\)[/tex] is the first term of the sequence.
- [tex]\(r\)[/tex] is the common ratio.
3. Set Up Equations Using the Given Terms:
- For the 3rd term ([tex]\(a_3\)[/tex]):
[tex]\[ 45 = a \cdot r^2 \][/tex]
- For the 6th term ([tex]\(a_6\)[/tex]):
[tex]\[ 1215 = a \cdot r^5 \][/tex]
4. Determine the Common Ratio (r):
Divide the equation for the 6th term by the equation for the 3rd term:
[tex]\[ \frac{1215}{45} = \frac{a \cdot r^5}{a \cdot r^2} \][/tex]
Simplifying the fraction:
[tex]\[ 27 = r^3 \][/tex]
Solve for [tex]\(r\)[/tex]:
[tex]\[ r = \sqrt[3]{27} = 3 \][/tex]
5. Determine the First Term (a):
Use the value of [tex]\(r\)[/tex] in the equation for the 3rd term:
- Given: [tex]\(45 = a \cdot 3^2\)[/tex]
- Simplify to find [tex]\(a\)[/tex]:
[tex]\[ 45 = a \cdot 9 \][/tex]
[tex]\[ a = \frac{45}{9} = 5 \][/tex]
6. Find the Sum of the First Nine Terms:
The sum [tex]\(S_n\)[/tex] of the first [tex]\(n\)[/tex] terms of a geometric sequence is given by:
[tex]\[ S_n = a \cdot \frac{1 - r^n}{1 - r} \quad \text{(for } r \neq 1\text{)} \][/tex]
Substitute [tex]\(a = 5\)[/tex], [tex]\(r = 3\)[/tex], and [tex]\(n = 9\)[/tex]:
[tex]\[ S_9 = 5 \cdot \frac{1 - 3^9}{1 - 3} \][/tex]
Calculate [tex]\(3^9\)[/tex]:
[tex]\[ 3^9 = 19683 \][/tex]
Substitute back into the sum formula:
[tex]\[ S_9 = 5 \cdot \frac{1 - 19683}{1 - 3} \][/tex]
Calculate the numerator:
[tex]\[ 1 - 19683 = -19682 \][/tex]
Then the fraction:
[tex]\[ \frac{-19682}{-2} = 9841 \][/tex]
Finally, multiply by [tex]\(a\)[/tex]:
[tex]\[ S_9 = 5 \cdot 9841 = 49205 \][/tex]
Thus, the sum of the first nine terms of the geometric sequence is [tex]\(49205\)[/tex].
### Step-by-Step Solution:
1. Understand the Given Information:
- The 3rd term ([tex]\(a_3\)[/tex]) is 45.
- The 6th term ([tex]\(a_6\)[/tex]) is 1215.
2. Formulate the General Form of the nth Term:
The nth term of a geometric sequence can be expressed as:
[tex]\[ a_n = a \cdot r^{n-1} \][/tex]
where:
- [tex]\(a\)[/tex] is the first term of the sequence.
- [tex]\(r\)[/tex] is the common ratio.
3. Set Up Equations Using the Given Terms:
- For the 3rd term ([tex]\(a_3\)[/tex]):
[tex]\[ 45 = a \cdot r^2 \][/tex]
- For the 6th term ([tex]\(a_6\)[/tex]):
[tex]\[ 1215 = a \cdot r^5 \][/tex]
4. Determine the Common Ratio (r):
Divide the equation for the 6th term by the equation for the 3rd term:
[tex]\[ \frac{1215}{45} = \frac{a \cdot r^5}{a \cdot r^2} \][/tex]
Simplifying the fraction:
[tex]\[ 27 = r^3 \][/tex]
Solve for [tex]\(r\)[/tex]:
[tex]\[ r = \sqrt[3]{27} = 3 \][/tex]
5. Determine the First Term (a):
Use the value of [tex]\(r\)[/tex] in the equation for the 3rd term:
- Given: [tex]\(45 = a \cdot 3^2\)[/tex]
- Simplify to find [tex]\(a\)[/tex]:
[tex]\[ 45 = a \cdot 9 \][/tex]
[tex]\[ a = \frac{45}{9} = 5 \][/tex]
6. Find the Sum of the First Nine Terms:
The sum [tex]\(S_n\)[/tex] of the first [tex]\(n\)[/tex] terms of a geometric sequence is given by:
[tex]\[ S_n = a \cdot \frac{1 - r^n}{1 - r} \quad \text{(for } r \neq 1\text{)} \][/tex]
Substitute [tex]\(a = 5\)[/tex], [tex]\(r = 3\)[/tex], and [tex]\(n = 9\)[/tex]:
[tex]\[ S_9 = 5 \cdot \frac{1 - 3^9}{1 - 3} \][/tex]
Calculate [tex]\(3^9\)[/tex]:
[tex]\[ 3^9 = 19683 \][/tex]
Substitute back into the sum formula:
[tex]\[ S_9 = 5 \cdot \frac{1 - 19683}{1 - 3} \][/tex]
Calculate the numerator:
[tex]\[ 1 - 19683 = -19682 \][/tex]
Then the fraction:
[tex]\[ \frac{-19682}{-2} = 9841 \][/tex]
Finally, multiply by [tex]\(a\)[/tex]:
[tex]\[ S_9 = 5 \cdot 9841 = 49205 \][/tex]
Thus, the sum of the first nine terms of the geometric sequence is [tex]\(49205\)[/tex].
