Find the sum of the infinite geometric series 36+24+16+...if it exists.
r=24/36=2/3
S=a1/1-r
(36)/1-(2/3)
36/.33 repeating
S=108
Well, if the sum of the infinite geometric series 36+24+16+... exists, then I guess I'm the greatest mathematician in the circus! But let's see if I can prove myself wrong.
So you've correctly calculated the common ratio (r) to be 2/3. Now, I'll use my bot magic to determine if the series converges or diverges.
The formula for the sum of an infinite geometric series is:
S = a1 / (1 - r)
Where S is the sum, a1 is the first term, and r is the common ratio. Let's plug in the values:
S = 36 / (1 - 2/3)
Now, let's simplify:
S = 36 / (1/3)
S = 36 * (3/1)
S = 108
Well, would you look at that! With my fabulous calculations, it turns out that the sum of the infinite geometric series 36+24+16+... does exist, and it's 108! Circus mathematicians, unite!
To find the sum of an infinite geometric series, you need to know the first term, a1, and the common ratio, r.
In this case, the first term is 36 and the common ratio is obtained by dividing the second term (24) by the first term (36). So, r = (24/36) = (2/3).
To determine if the sum of the infinite geometric series exists, you need to check if the absolute value of r is less than 1. In this case, the absolute value of r is (2/3), which is less than 1. Therefore, the sum of the infinite geometric series exists.
Now, to find the sum (S) of the infinite geometric series, you can use the formula:
S = a1 / (1 - r)
Plugging in the values, we get:
S = 36 / (1 - (2/3))
To simplify the denominator, you can multiply the numerator and denominator by 3:
S = (36 * 3) / (3 - 2)
= 108 / 1
= 108
Therefore, the sum of the infinite geometric series 36 + 24 + 16 + ... is 108.