Find the sum of the first six terms of the geometric series 81 -27 +9 -...
how do you work it??
Well, first of all, let's appreciate the brilliance of this series. It's like a math magician playing tricks on us!
Now, let's break this down. We have -27 divided by 81, which is -1/3. Then we have 9 divided by -27, which is -1/3. Do you see a pattern here? It's like we're dividing by -3 at every step!
So, we can say that the common ratio (r) of this geometric series is -1/3.
Now, to find the sum of the first six terms, we can use the formula for the sum of a geometric series:
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, a = 81, r = -1/3, and n = 6.
Plugging in the values, we get:
S = 81 * (1 - (-1/3)^6) / (1 - (-1/3)).
Now all we have to do is calculate this expression and we will have our answer!
To find the sum of the first six terms of a geometric series, you need to use the formula for the sum of a geometric series.
The formula for the sum of a geometric series is:
S = a * (1 - r^n) / (1 - r)
where:
S is the sum of the series
a is the first term of the series
r is the common ratio of the series
n is the number of terms in the series
In the given series, the first term (a) is 81, and the common ratio (r) is -27 / 81 = -1/3. The number of terms (n) is 6.
Using the formula, plug in the values and calculate:
S = 81 * (1 - (-1/3)^6) / (1 - (-1/3))
Simplifying the calculations:
S = 81 * (1 - (1/729)) / (1 + 1/3)
S = 81 * (1 - 1/729) / (4/3)
S = 81 * (728/729) / (4/3)
S = (81 * 728) / ((4/3) * 729)
S = (59088) / (1.33)
S ≈ 44388.72
Therefore, the sum of the first six terms of the given geometric series is approximately 44388.72.
To find the sum of the first six terms of a geometric series, we can use the formula:
S = a(1 - r^n) / (1 - r),
where:
S = sum of the series,
a = first term of the series,
r = common ratio,
n = number of terms.
In this case, the first term (a) is 81 and the common ratio (r) is -27/81 = -1/3. Since we want to find the sum of the first six terms (n = 6), we can substitute these values into the formula and calculate the sum.
S = 81 * (1 - (-1/3)^6) / (1 - (-1/3))
Now, let's solve this:
S = 81 * (1 - 1/729) / (1 + 1/3)
= 81 * (728/729) / (4/3),
= 81 * 728/729 * 3/4.
Now, we can simplify and calculate the sum:
S = (81 * 728 * 3) / (729 * 4),
= 157464 / 2916,
= 54.
Therefore, the sum of the first six terms of the given geometric series is 54.
just add up the 6 terms , it is probably easier than using the formula
81 - 27 + 9 - 3 + 1 - 1/3
= 182/3 or 60 2/3 or 60.6666..