A new annual festival In the first year, the festival had 25 participants. It is expected that the number of people attending the festival will double every year.
5 years ago
Answered By Kiley C
Hi There!
So in order to solve this question as a geometric series, you need to also have a number of years (which is missing above). Geometric series are related to geometric sequences and that will somewhat make it easier to understand. A Geometric Sequence is just the sequence of numbers where each term after the first is found by multiplying the previous term by a fixed ratio. In this question, our ratio is 2. Therefore, the geometric sequence would look like this (if you think of each term as a year):
Then to find the Geometric Series - we sum the terms in the geometric sequence. Our series will be in the following form:
Sn = a + (a * r) + (a * r2) + ... + (a * r n-1) where "a" is our first term and "n" is the number of terms in our sequence. This simplifies down to the equation: Sn = $\frac{a\left(1-r^n\right)}{1-r}$a(1−rn)1−r
Now considering the 4 terms we have above (meaning the festival lasted for 4 years), the 2 ways to find the series are:
or Sn = $\frac{25\left(1-2^4\right)}{1-2}$25(1−24)1−2 = $\frac{25\left(1-16\right)}{-1}$25(1−16)−1 = $\frac{25\left(-15\right)}{-1}$25(−15)−1 = $\frac{-375}{-1}$−375−1 = 375 So 375 people would be the total attendees that attended the festival over 4 years. Hope that helps!
5 years ago
Answered By Kiley C
Hi There!
So in order to solve this question as a geometric series, you need to also have a number of years (which is missing above). Geometric series are related to geometric sequences and that will somewhat make it easier to understand. A Geometric Sequence is just the sequence of numbers where each term after the first is found by multiplying the previous term by a fixed ratio. In this question, our ratio is 2. Therefore, the geometric sequence would look like this (if you think of each term as a year):
t1 = a * r0 = 25 * 20 = 25 t2 = a * r1 = 25 * 21 = 25 * 2 = 50t3 = a * r2 = 25 * 22 = 25 * 2 * 2 = 50 * 2 = 100t4 = a * r3 = 25 * 23 = 25 * 2 * 2 * 2 = 50 * 2 * 2 = 100 * 2 = 200
Then to find the Geometric Series - we sum the terms in the geometric sequence. Our series will be in the following form:
Sn = a + (a * r) + (a * r2) + ... + (a * r n-1) where "a" is our first term and "n" is the number of terms in our sequence. This simplifies down to the equation: Sn = $\frac{a\left(1-r^n\right)}{1-r}$a(1−rn)1−r
Now considering the 4 terms we have above (meaning the festival lasted for 4 years), the 2 ways to find the series are:
Sn = 25 + (25 * 2) + (25 * 22) + (25 * 2(4-1) ) = 25 + 50 + (25 * 4) + (25 * 23) = 25 + 50 + 100 + (25 * 8) = 25 + 50 + 100 + 200 = 375
or Sn = $\frac{25\left(1-2^4\right)}{1-2}$25(1−24)1−2 = $\frac{25\left(1-16\right)}{-1}$25(1−16)−1 = $\frac{25\left(-15\right)}{-1}$25(−15)−1 = $\frac{-375}{-1}$−375−1 = 375 So 375 people would be the total attendees that attended the festival over 4 years. Hope that helps!