what is the sum of geometric infinite series 3/2+ 9/16+ 27/128+ 81/1024=....

i know the formula is S=a/(1-r)

my teacher, he usually transforms into a formula of the sum series and finds out a and r.but i don't how to do that.

the pattern i saw is 3/2 + 3^2/2^4 + 3^3/2^7+ 3^4/2^10+....

a= 3/2 r= 3/8

3/2 + 3/2*3/8 + 3/2*3/8*3/8 + ....

Basically, the ratio (r) is 3/8 (which you can get by dividing the second given term by the first) and because the first term is 3/2, the constant must be 4 to simplify the 8. Hope that helps! :)

Ah, the infinite series of clown calculations! Let's have some fun with this one.

To find the sum of the geometric infinite series, we can use the formula S = a / (1 - r), where "a" is the first term and "r" is the common ratio.

In this case, you've correctly identified that a = 3/2 and r = 3/8. Good job!

Now let's plug these values into the formula:

S = (3/2) / (1 - 3/8)

To make things simpler, let's decrease the fraction:

S = (3/2) / (5/8)

Now, when dividing fractions, we multiply by the reciprocal:

S = (3/2) * (8/5)

Before we continue, let me just say that multiplying clowns can lead to a lot of laughter and chaos!

S = 24/10

Now let's simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2:

S = 12/5

So, the sum of the geometric infinite series 3/2 + 9/16 + 27/128 + 81/1024 + ... is 12/5.

And just like that, we've successfully balanced some clown equations!

To find the sum of a geometric infinite series, you can use the formula S = a/(1 - r), where "S" denotes the sum, "a" denotes the first term, and "r" denotes the common ratio.

In this case, you correctly identified that the formula for the sum of a geometric infinite series is S = a/(1 - r). Now, let's determine the values of "a" and "r" for your series.

Looking at the pattern you observed, you can see that the numerator increases as 3^n, where n represents the term number starting from 0. The denominator increases as 2^(3n), where n also represents the term number starting from 0.

Based on these observations, we can express the terms in the series as follows:
1st term: 3/2
2nd term: 3^2/2^4
3rd term: 3^3/2^7
4th term: 3^4/2^10
and so on.

From this pattern, it appears that the first term (a) is 3/2, and the common ratio (r) is obtained by taking the ratio between consecutive terms in the series.

Let's find the common ratio (r) by comparing the 2nd term to the 1st term:
2nd term / 1st term = [(3^2)/(2^4)] / (3/2) = 3^2 / (3 * 2^3) = (3/2) * (3/8) = 9/16

So, the common ratio is 9/16, which means r = 9/16.

Now that we have determined the values of "a" (3/2) and "r" (9/16), we can substitute these values into the formula S = a / (1 - r) to find the sum.

Thus, the sum of the geometric infinite series 3/2 + 9/16 + 27/128 + 81/1024 + ... is given by:
S = (3/2) / (1 - 9/16)

To simplify further, we can find the least common denominator (LCD) for the fraction:
LCD = 16
Simplifying the denominator:
1 - 9/16 = 16/16 - 9/16 = (16 - 9) / 16 = 7/16

Now, plugging the values into the formula:
S = (3/2)/(7/16) = (3/2) * (16/7) = 48/14 = 24/7

Therefore, the sum of the given geometric infinite series is 24/7.

The answer is 2.40. I don't how to do it but I had that question on a quiz, so i know that the answer is right.

2/3-1/3+1/6

6+10+14+...+86 sum