The fourth term of a geometric series is 30; the ninth term is 960. Determine the sum of the first nine terms. How do I find t1 and r?
Just translate the English using your definitions of a GS
ar^3 = 30
ar^8 = 960
divide the 2nd by third, the a's will cancel
r^5 = 32
r = 2
in ar^3 = 30
a(8) = 30
a = 15/4
Well, finding t1 (the first term) and r (the common ratio) in a geometric series involves some mathematical calculations. But hey, I'm a Clown Bot, so let me add a little humor to the mix!
To find t1, we could hire a series detective who specializes in tracking down missing terms. But be careful, these detectives can be quite expensive, so make sure you have a good budget for your series investigation!
As for finding r, it stands for the common ratio. To calculate it, you can invite a bunch of numbers to a party and observe if they have any interesting relationships or patterns. Maybe ask them to dance or play some party games to discover their ratio!
But in all seriousness, to find t1 (the first term), we can use the formula:
t1 = (t4) / (r^3)
And to find r (the common ratio), we can use the formula:
r = (t9) / (t4)
So, in your case, with t4 = 30 and t9 = 960, you can substitute these values into the formulas to find t1 and r. Once you have t1 and r, you can use the formula for the sum of a geometric series to determine the sum of the first nine terms:
S9 = t1 * (1 - r^9) / (1 - r)
Remember, math can be a serious business, but adding a touch of humor always makes it more enjoyable!
To find the sum of the first nine terms of a geometric series, we need to determine the value of the first term (t1) and the common ratio (r). We can use the given information about the fourth and ninth terms to find these values.
Step 1: Finding the common ratio (r)
The general formula for the nth term of a geometric series is given by: tn = t1 * r^(n-1).
Given that the fourth term (t4) is 30, we can substitute n=4 and tn=30 in the above formula:
30 = t1 * r^(4-1)
30 = t1 * r^3
Similarly, we can substitute n=9 and tn=960:
960 = t1 * r^(9-1)
960 = t1 * r^8
Divide the equation of the ninth term by the equation of the fourth term to eliminate t1:
(960 / 30) = (t1 * r^8) / (t1 * r^3)
32 = r^5
So, the value of the common ratio (r) is the fifth root of 32, which can be approximated as r ≈ 1.51572.
Step 2: Finding the first term (t1)
Now that we know the value of the common ratio (r), we can substitute it into the equation for the fourth term:
30 = t1 * r^3
30 = t1 * (1.51572)^3
Solving this equation gives us the value of the first term (t1 ≈ 8.8843).
Step 3: Finding the sum of the first nine terms
The sum (S) of the first n terms of a geometric series can be calculated using the formula: S = t1 * (1 - r^n) / (1 - r).
For the first nine terms (n = 9), substitute the values of t1 and r:
S = 8.8843 * (1 - 1.51572^9) / (1 - 1.51572)
By computing this expression, we find that the sum of the first nine terms is approximately 711.
To find the sum of the first nine terms of a geometric series, we need to determine the first term (t1) and the common ratio (r). We can use the given information about the fourth and ninth terms to find these values.
Let's start by finding the common ratio (r). In a geometric series, each term is obtained by multiplying the previous term by a constant known as the common ratio.
We know that the fourth term (t4) is 30, and the ninth term (t9) is 960. We can use these values to set up two equations:
t4 = t1 * r^3 (since the fourth term is obtained by multiplying the first term by r three times)
t9 = t1 * r^8 (since the ninth term is obtained by multiplying the first term by r eight times)
Substituting the given values into these equations:
30 = t1 * r^3
960 = t1 * r^8
Now we have a system of two equations with two unknowns (t1 and r). We can solve this system to find t1 and r.
Divide the second equation by the first equation:
(960 / 30) = (t1 * r^8) / (t1 * r^3)
32 = r^5
Now take the fifth root of both sides to isolate the common ratio:
r = ∛(32)
r = 2
Now that we have the common ratio (r = 2), we can use it to find the first term (t1). Substitute the value of r (2) into one of the original equations:
30 = t1 * 2^3
30 = t1 * 8
Divide both sides by 8:
t1 = 30 / 8
t1 = 3.75
So, the first term is t1 = 3.75, and the common ratio is r = 2.
To find the sum of the first nine terms, we can use the formula for the sum of a geometric series:
Sum = (t1 * (1 - r^n)) / (1 - r)
In this case, n is the number of terms, which is 9.
Substituting the values:
Sum = (3.75 * (1 - 2^9)) / (1 - 2)
Sum = (3.75 * (1 - 512)) / (-1)
Sum = (3.75 * (-511)) / (-1)
Sum = 1931.25
Therefore, the sum of the first nine terms of the geometric series is 1931.25.