Sums of Geometric Series

This means that a common ratio exists between the number of newly-infected students each day. Therefore, their sum represents a geometric series. With this in mind, an explicit rule can be written to find the number of newly-infected students on the n^\text{th} day. a_n=a_1 * r^(n-1) Since only Tearrik fell ill on the first day, the first term of the sequence is a_1= 1 and the common ratio r= 3. Substitute these values into the formula to find the explicit rule for this sequence. The number of newly-infected students on the n^\text{th} day can be represented by the formula a_n=3^(n-1). S_n=a_1(1-r^n)/1-r Since the total number of infected students on the 7^\text{th} day is required, n= 7 will be substituted into the formula. Remember that the first term a_1 is 1 and the common ratio r is 3. If the school does not suspend classes, there will be 1093 infected students in 7 days.

The given sum is a finite geometric series. Therefore, the formula for the sum of a finite geometric series can be used to find this sum. S_n=a_1(1-r^n)/1-r , r ≠ 1 In this formula, a_1 is the first term, r is the common ratio, and n is the number of terms. First, substitute i=1 into the expression given on the right side of the sigma notation to find a_1. The first term is 18. Notice that the summation index i is placed as a single exponent of 3, meaning that the next term can be found by multiplying the previous one by 3. This makes the common ratio r=3. Another way to find r is to divide the second term by the first. Start by substituting i=2 into the given expression. Now calculate the ratio of a_2 to a_1. a_2/a_1=54/18 ⇒ r=3 The common ratio was found to be 3. Since the lower limit of the sigma notation is 1 and the upper limit of the sigma notation is 12, there are 12 terms in the series in total, so n= 12. The sum can be calculated with these values. To do so, substitute all the values into the formula for S_n and simplify.

Initial Height:& 3 m [0.5em] First Bounce:& 3 (0.85) m For the second bounce, the height of the ball will be 0.85 times the first bounce, 3 (0.85) meters. Second Bounce: 3 (0.85)(0.85) ⇒   3 (0.85)^2 m The heights of the other bounces can be written by considering this pattern.

Because the ball rises and then falls the same distance after each bounce, the vertical distance traveled is 2 times 0.85 of the previous bounce. Since the initial height of the ball is 3 meters, the total distance traveled by the ball can be written as follows. 3 + 2 * 3 (0.85) + 2 * 3 (0.85)^2 + ⋯ ⇓ 3 + 6 (0.85) +6 (0.85)^2 + ⋯ Since a common ratio r= 0.85 exists between the distances that the ball travels, the distances traveled starting with the first bounce represent a geometric series. This sum is considered infinite because it is assumed that the ball could continue to bounce in increasingly small increments forever. Now, express the sum using summation notation. 3+∑_(n=1)^(∞)6(0.85)^n Note that even though the sum of the series is infinite, it can still be found by using the formula for the sum of an infinite series because the absolute value of the common ratio |r|=0.85 is less than 1. S_(∞)=a_1/1- r The first term of the infinite geometric series is 6(0.85) and the common ratio is 0.85. Now, substitute these values into the formula! ∑_(n=1)^(∞)6(0.85)^n = 6(0.85)/1- 0.85 Then, evaluate the right hand side. The sum of the series is 34. Now the initial 3-meter height of the ball will be added. 34 + 3 = 37 This means that the ball will have traveled 37 meters when it comes to rest. 3 + 2 * 3 (0.85) + 2 * 3(0.85)^2 + 2 * 3 (0.85)^3 ⇓ 3 + 6(0.85) +6(0.85)^2 +6(0.85)^3 Notice that this sum represents a partial sum of the infinite series considered in Part A. To calculate this partial sum, the formula for the sum of a finite geometric series will be used, as the sum is calculated up until n= 3. S_n=a_1(1- r^n)/1- r With this in mind, substitute n=3, a_1=6(0.85), and r=0.85 into the formula and evaluate the result. Now, add the fourth vertical distance 3(0.85)^4 to this sum. Finally, the initial 3-meter height will be added. 14.69+3=17.69 The ball will travel vertically about 17.69 meters in total until Tadeo catches it at the top of the fourth bounce.

To determine who will save more money in 15 months, the total amounts saved will be calculated one at a time.

Tadeo's Savings

Tadeo is planning to save 100 dollars every month. This means that his savings will increase linearly by a constant rate of 100 dollars. Therefore, multiply 100 dollars by 15 months to find the total amount of money he will have saved in 15 months. 100 * 15 = 1500 After 15 months, Tadeo will have saved 1500 dollars.

Tearrik's Savings

Tearrik will start by saving only $ 0.50 the first month. Then, he will increase the amount of money he saves every month by doubling the previous amount.

Since there is a common ratio between the amount of money saved each month, this sum represents a geometric series. Therefore, the formula for the sum of a finite geometric series will be used to calculate the total amount of money Tearrik will have saved in 15 months. S_n=a_1(1- r^n)/1- r Now, substitute n= 15, a_1= 0.50, and r= 2 into this formula and simplify. After 15 months, Tearrik will have saved 16 383.5 dollars.

Conclusion

Notice that Tearrik's savings will increase rapidly although he starts with an extremely small amount. On the other hand, Tadeo will save the same amount of money each month and his savings will therefore increase at a constant rate. Finally, after 15 months, Tearrik will have saved much more money than Tadeo.