Answer :
To find the sum of the geometric sequence [tex]\(-4, 24, -144, \ldots\)[/tex] if there are 6 terms, we can follow these steps:
1. Identify the First Term and Common Ratio:
- The first term of the sequence ([tex]\(a_1\)[/tex]) is [tex]\(-4\)[/tex].
- To find the common ratio ([tex]\(r\)[/tex]), we divide the second term by the first term:
[tex]\[
r = \frac{24}{-4} = -6
\][/tex]
2. Use the Formula for the Sum of a Geometric Sequence:
- The formula to find the sum [tex]\(S_n\)[/tex] of the first [tex]\(n\)[/tex] terms of a geometric sequence is:
[tex]\[
S_n = a_1 \frac{1 - r^n}{1 - r}
\][/tex]
- Here, [tex]\(n = 6\)[/tex], [tex]\(a_1 = -4\)[/tex], and [tex]\(r = -6\)[/tex].
3. Plug in the Values:
- Substitute the values into the formula:
[tex]\[
S_6 = -4 \times \frac{1 - (-6)^6}{1 - (-6)}
\][/tex]
4. Calculate the Value Inside the Formula:
- First, calculate [tex]\((-6)^6\)[/tex]:
[tex]\[
(-6)^6 = 46656
\][/tex]
- Substitute back into the formula:
[tex]\[
S_6 = -4 \times \frac{1 - 46656}{1 + 6}
\][/tex]
5. Simplify the Expression:
- Simplify inside the fraction:
[tex]\[
S_6 = -4 \times \frac{1 - 46656}{7}
\][/tex]
- Simplify further:
[tex]\[
S_6 = -4 \times \frac{-46655}{7}
\][/tex]
- Calculate the division:
[tex]\[
S_6 = -4 \times (-6665)
\][/tex]
6. Find the Final Sum:
- Multiply to get the sum:
[tex]\[
S_6 = 26660
\][/tex]
The sum of the geometric sequence is 26,660.
1. Identify the First Term and Common Ratio:
- The first term of the sequence ([tex]\(a_1\)[/tex]) is [tex]\(-4\)[/tex].
- To find the common ratio ([tex]\(r\)[/tex]), we divide the second term by the first term:
[tex]\[
r = \frac{24}{-4} = -6
\][/tex]
2. Use the Formula for the Sum of a Geometric Sequence:
- The formula to find the sum [tex]\(S_n\)[/tex] of the first [tex]\(n\)[/tex] terms of a geometric sequence is:
[tex]\[
S_n = a_1 \frac{1 - r^n}{1 - r}
\][/tex]
- Here, [tex]\(n = 6\)[/tex], [tex]\(a_1 = -4\)[/tex], and [tex]\(r = -6\)[/tex].
3. Plug in the Values:
- Substitute the values into the formula:
[tex]\[
S_6 = -4 \times \frac{1 - (-6)^6}{1 - (-6)}
\][/tex]
4. Calculate the Value Inside the Formula:
- First, calculate [tex]\((-6)^6\)[/tex]:
[tex]\[
(-6)^6 = 46656
\][/tex]
- Substitute back into the formula:
[tex]\[
S_6 = -4 \times \frac{1 - 46656}{1 + 6}
\][/tex]
5. Simplify the Expression:
- Simplify inside the fraction:
[tex]\[
S_6 = -4 \times \frac{1 - 46656}{7}
\][/tex]
- Simplify further:
[tex]\[
S_6 = -4 \times \frac{-46655}{7}
\][/tex]
- Calculate the division:
[tex]\[
S_6 = -4 \times (-6665)
\][/tex]
6. Find the Final Sum:
- Multiply to get the sum:
[tex]\[
S_6 = 26660
\][/tex]
The sum of the geometric sequence is 26,660.
