Answer :
We are given a geometric sequence with the first term
[tex]$$a=-4,$$[/tex]
common ratio
[tex]$$r=-6,$$[/tex]
and the number of terms
[tex]$$n=6.$$[/tex]
The sum of the first [tex]$n$[/tex] terms of a geometric sequence is given by
[tex]$$
S_n = a \times \frac{r^n - 1}{r - 1}.
$$[/tex]
Step 1. Compute [tex]$r^n$[/tex]:
We calculate
[tex]$$
r^6 = (-6)^6.
$$[/tex]
Since an even power of a negative number is positive, we have
[tex]$$
(-6)^6 = 46656.
$$[/tex]
Step 2. Compute the numerator of the formula:
Subtract 1 from [tex]$r^6$[/tex]:
[tex]$$
r^6 - 1 = 46656 - 1 = 46655.
$$[/tex]
Step 3. Compute the denominator of the formula:
Subtract 1 from [tex]$r$[/tex]:
[tex]$$
r - 1 = -6 - 1 = -7.
$$[/tex]
Step 4. Substitute into the sum formula:
Now replace [tex]$a$[/tex], the numerator, and the denominator in the formula:
[tex]$$
S_6 = -4 \times \frac{46655}{-7}.
$$[/tex]
The negatives cancel out:
[tex]$$
S_6 = \frac{-4 \times 46655}{-7} = 4 \times \frac{46655}{7}.
$$[/tex]
Step 5. Simplify to obtain the final sum:
Multiplying and dividing gives
[tex]$$
S_6 = 26660.
$$[/tex]
Thus, the sum of the geometric sequence with 6 terms is
[tex]$$\boxed{26660}.$$[/tex]
[tex]$$a=-4,$$[/tex]
common ratio
[tex]$$r=-6,$$[/tex]
and the number of terms
[tex]$$n=6.$$[/tex]
The sum of the first [tex]$n$[/tex] terms of a geometric sequence is given by
[tex]$$
S_n = a \times \frac{r^n - 1}{r - 1}.
$$[/tex]
Step 1. Compute [tex]$r^n$[/tex]:
We calculate
[tex]$$
r^6 = (-6)^6.
$$[/tex]
Since an even power of a negative number is positive, we have
[tex]$$
(-6)^6 = 46656.
$$[/tex]
Step 2. Compute the numerator of the formula:
Subtract 1 from [tex]$r^6$[/tex]:
[tex]$$
r^6 - 1 = 46656 - 1 = 46655.
$$[/tex]
Step 3. Compute the denominator of the formula:
Subtract 1 from [tex]$r$[/tex]:
[tex]$$
r - 1 = -6 - 1 = -7.
$$[/tex]
Step 4. Substitute into the sum formula:
Now replace [tex]$a$[/tex], the numerator, and the denominator in the formula:
[tex]$$
S_6 = -4 \times \frac{46655}{-7}.
$$[/tex]
The negatives cancel out:
[tex]$$
S_6 = \frac{-4 \times 46655}{-7} = 4 \times \frac{46655}{7}.
$$[/tex]
Step 5. Simplify to obtain the final sum:
Multiplying and dividing gives
[tex]$$
S_6 = 26660.
$$[/tex]
Thus, the sum of the geometric sequence with 6 terms is
[tex]$$\boxed{26660}.$$[/tex]
