To find the sixth term of a geometric sequence given the first term [tex]\(a_1\)[/tex] and the common ratio [tex]\(r\)[/tex], we can use the formula for the [tex]\(n\)[/tex]-th term of a geometric sequence:
[tex]\[ a_n = a_1 \cdot r^{n-1} \][/tex]
Here's a step-by-step approach to finding the sixth term:
1. Determine the values:
- The first term [tex]\(a_1\)[/tex] is given as [tex]\(5\)[/tex].
- The common ratio [tex]\(r\)[/tex] is given as [tex]\(3\)[/tex].
- We want to find the sixth term, so [tex]\(n = 6\)[/tex].
2. Plug in the values into the formula:
[tex]\[ a_6 = a_1 \cdot r^{6-1} \][/tex]
[tex]\[ a_6 = 5 \cdot 3^{5} \][/tex]
3. Calculate the exponent:
[tex]\[ 3^5 \][/tex]
This means 3 multiplied by itself 5 times:
[tex]\[ 3 \times 3 \times 3 \times 3 \times 3 = 243 \][/tex]
4. Multiply the first term by the result of the exponentiation:
[tex]\[ a_6 = 5 \cdot 243 \][/tex]
5. Perform the multiplication:
[tex]\[ 5 \cdot 243 = 1215 \][/tex]
Therefore, the sixth term of the geometric sequence is [tex]\(1215\)[/tex].
