Q: 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.
^Are there two answers, since the common ratio could be positive or negative?
To determine the sum of the first seven terms of a geometric series, we need to find the common ratio and then use the formula for the sum of a geometric series.
Given that the first term (a1) is 11 and the seventh term (a7) is 704, we can set up the following equations:
a1 = 11
a7 = 704
The general formula for the nth term of a geometric series is given by:
an = a1 * r^(n-1)
Substituting the values we have, we can set up the equations:
11 * r^(7-1) = 704
Simplifying the equation, we get:
11 * r^6 = 704
Now, let's find the common ratio (r):
Divide both sides of the equation by 11:
r^6 = 64
Taking the 6th root of both sides:
r = 2
Since the common ratio is positive (r = 2), there is only one answer.
To find the sum of the first seven terms (S7) using the formula for the sum of a geometric series, we can use the equation:
S7 = a1 * (1 - r^7) / (1 - r)
Substituting the values we have:
S7 = 11 * (1 - 2^7) / (1 - 2)
Simplifying the equation:
S7 = 11 * (1 - 128) / (1 - 2)
S7 = 11 * (-127) / (-1)
S7 = 11 * 127
S7 = 1397
Therefore, the sum of the first seven terms in the geometric series is 1397.
To find the sum of the first seven terms of a geometric series, we can use the formula:
S = a * (r^n - 1) / (r - 1)
Where:
S is the sum of the geometric series,
a is the first term of the series,
r is the common ratio,
and n is the number of terms.
In this case, we are given that the first term (a) is 11 and the seventh term is 704.
To find the common ratio (r), we can use the formula for the nth term of a geometric series:
An = a * (r^(n-1))
Where An is the nth term of the series. Plugging in the values we have, we get:
704 = 11 * (r^(7 - 1))
Simplifying, we have:
704 = 11 * r^6
Divide both sides by 11:
64 = r^6
Taking the sixth root of both sides, we get:
r = ±2
Since the common ratio could be positive or negative, there are two possible answers to the sum of the first seven terms of the geometric series. We'll calculate both:
Using r = 2:
S = 11 * (2^7 - 1) / (2 - 1)
= 11 * (128 - 1) / 1
= 11 * 127
= 1,397
Using r = -2:
S = 11 * ((-2)^7 - 1) / (-2 - 1)
= 11 * (-128 - 1) / -3
= 11 * (-129) / -3
= 477
Therefore, the sum of the first seven terms of the geometric series could be 1,397 or 477, depending on whether the common ratio is positive or negative.
The ratio between successive terms is
(704/11)^(1/6) = 2 (or -2)
Yes, two answers are possible.
One series would be
11, 22, 44, 88, 176, 352, 704
and the other
11, -22, 44, -88, 176, -352, 704
You do the adding
Well, let's calculate the common ratio first. We know that the seventh term (704) divided by the first term (11) should give us the common ratio. So, 704 divided by 11 is 64.
Now, to find the sum of the first seven terms, we can use the formula: 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 is 11, r is 64, and n is 7.
Plugging in the values, we get:
S = 11 * (1 - 64^7) / (1 - 64)
Now, let's calculate that...
Oops, I forgot to carry my calculator with me! My bad! But hey, at least we found the common ratio!