Determine the sum of the first seven terms of a geometric series in which the first term is 11 and the seventh term is 704.
a = 11
ar^6 = 704 = 11*64
so, r^6 = 64, meaning r = ±2
As usual, the sum of n terms is
Sn = a(r^n-1)/(r-1)
Well, let's see here. The first term of the geometric series is 11, and the seventh term is 704. That's quite a jump! It's like going from playing a mini-golf to suddenly playing in the Olympics.
To find the sum of the first seven terms, we need to figure out the common ratio of the series. But let's not be too hasty. I don't want to be blamed for causing any math accidents.
If we divide the seventh term by the first term, we get 704 divided by 11, which equals 64. So, the common ratio of this geometric series is 64. That's a pretty big jump from term to term!
Now, to find the sum of the first seven terms, we can use the formula:
sum = (first term * (1 - common ratio^n)) / (1 - common ratio)
Here, n represents the number of terms we want to sum. In this case, it's 7.
So, plugging in the numbers, we get:
sum = (11 * (1 - 64^7)) / (1 - 64)
But honestly, this formula is starting to bore me a bit. It's like trying to explain a joke that nobody gets. So, let's just cut to the chase.
The sum of the first seven terms of this geometric series is 11 x (1 - 64^7) / (1 - 64). And I apologize if my attempt at humor didn't quite live up to your expectations. Sometimes jokes are like geometry – they just don't always add up.
To determine the sum of the first seven terms of a geometric series, we need to find the common ratio (r).
We are given that the first term (a₁) is 11 and the seventh term (a₇) is 704.
We can use the formula for the nth term of a geometric sequence:
aₙ = a₁ * r^(n-1)
Substituting the given values:
a₇ = 11 * r^(7-1)
704 = 11 * r^6
To find the common ratio (r), we can divide the equation by 11:
64 = r^6
Taking the 6th root of both sides, we find:
r = ±2
Since the common ratio cannot be negative, we take r = 2.
Now, we can use the formula for the sum of a finite geometric series:
Sn = a₁ * (1 - r^n) / (1 - r)
Substituting the given values:
S₇ = 11 * (1 - 2^7) / (1 - 2)
S₇ = 11 * (1 - 128) / (1 - 2)
S₇ = 11 * (-127) / (-1)
S₇ = 11 * 127
S₇ = 1397
Therefore, the sum of the first seven terms of the geometric series is 1397.
To determine the sum of the first seven terms of a geometric series, we need to find the common ratio (r) first. We are given the first term (a1) as 11 and the seventh term (a7) as 704.
The formula to find the nth term of a geometric series is: an = a1 * r^(n-1), where an is the nth term, a1 is the first term, r is the common ratio, and n is the term number.
Using a7 = 11 * r^(7-1) = 11 * r^6 = 704, we can solve for r:
11 * r^6 = 704
Divide both sides by 11:
r^6 = 704 / 11
r^6 ≈ 64
Taking the sixth root of both sides:
r ≈ ∛(64)
r ≈ 2
So, the common ratio (r) is approximately 2.
Now, we can use the formula for the sum of a geometric series to find the sum of the first seven terms. The formula is: Sn = a1 * (1 - r^n) / (1 - r), where Sn is the sum of the first n terms.
Substituting the given values, we have:
S7 = 11 * (1 - 2^7) / (1 - 2)
S7 = 11 * (1 - 128) / -1
S7 = 11 * (-127) / -1
S7 = 11 * 127
S7 = 1397
Therefore, the sum of the first seven terms of the geometric series is 1397.