The sum of the first n terms of the geometric sequence
-1, -3, -9,... is -3280. Find the value of n. Express answer in simplest fractional form.
1. t1= -1
2. r = 3
3. S6 = I got -364 but it's not the right answer
4. n = I keep getting a decimal number: 18.72075441...
a = -1
r = 3
Sn = (-1)(3^n-1)/(3-1)
= (1-3^n)/2 = -3280
1-3^n = -6560
3^n = 6561
n = 8
Just use the sum formula
for a GP
sum(n) = a(r^n - 1)/(r-1)
for ours: a = -1, r = 3
-1( 3^n - 1)/(3-1) = -3280
3^n - 1 = 6560
3^n = 6561
you could just play around with powers of 3
or you could use logs
n log3 = log 6561
n = log6561/log3 = 8
check:
sum(8) = -1(3^8 - 1)/(3-1)
= -1( 6561 - 1)/2
= -3280
To find the value of n, let's start by using the formula for the sum of a geometric sequence:
Sn = (a(1 - r^n))/(1 - r),
where Sn represents the sum of the first n terms, a is the first term, and r is the common ratio.
In this case, the first term a is -1 and the common ratio r is 3. We are given that the sum of the first n terms, Sn, is -3280.
Thus, we can set up the equation -3280 = (-1(1 - 3^n))/(1 - 3).
Simplifying this equation will help us find the value of n:
-3280 = (-1(1 - 3^n))/(-2),
Multiply both sides of the equation by -2:
6560 = 1 - 3^n.
Rearranging the equation:
3^n = 1 - 6560,
3^n = -6559.
Now, let's solve for n. Taking the logarithm base 3 of both sides:
n * log3(3) = log3(-6559),
n = log3(-6559).
However, logarithms of negative numbers are undefined in the real number system. Thus, there is no real solution for n in this case.
The sum of the first n terms of the given geometric sequence cannot be -3280. Please double-check the problem or provide additional information if needed.
To find the value of n for the sum of the first n terms of the geometric sequence -1, -3, -9,... given that the sum is -3280, we can use the formula for the sum of a geometric sequence:
S = a * (1 - r^n) / (1 - r),
where S is the sum, a is the first term, r is the common ratio, and n is the number of terms.
In this case, we have:
a = -1,
r = 3,
S = -3280.
Let's substitute these values into the formula and solve for n:
-3280 = -1 * (1 - 3^n) / (1 - 3).
Multiplying both sides of the equation by (1 - 3) to remove fractions, we get:
-3280 * (1 - 3) = -1 * (1 - 3^n).
Simplifying:
-3280 * (-2) = 1 - 3^n,
6560 = 1 - 3^n.
Now, let's solve for n.
Subtracting 1 from both sides:
6560 - 1 = -3^n,
6559 = -3^n.
To solve for n, we need to take the logarithm of both sides with base 3. This will allow us to isolate the exponent:
log3(6559) = log3(-3^n).
Using logarithm properties, we can bring the exponent down:
log3(6559) = n * log3(-3).
Since the logarithm of a negative number is not a real number, there is no real value of n that satisfies this equation. Therefore, there seems to be an error in the given information, as it is not possible to have a sum of -3280 for these terms of the geometric sequence.