You want to save $500 for a school trip. You begin by saving a penny on the first day. You save an additional penny each day after that. For example, you will save two pennies on the second day, three pennies on the third day, and so on.
a. How much money will you have saved after 100 days?
b. Use a series to determine how many days it takes you to save $500
+23245 On the first day you save $0.01, on the second day you save $0.02, on the third day you save $0.03, etc.
Thus, on the nth day, you save 0.01·n.
This is an arithmetic sequence because each number is 0.01 greater than the previous number.
The formula for the sum of an arithmetic sequence is: Sum = N(F + L)/2
where N = number of terms, F = value of the first term, L = value of the last term.
a) N = 100, F = 0.01, L = 0.01·N = 1.00 ---> Sum = 100(0.01 + 1.00)/2 ---> Sum = $50.50.
b) N = n, F = 0.01, L = 0.01n, Sum = 500.00 ---> 500.00 = n(0.01 + 0.01n)/2
---> 1000.00 = n(0.01 + 0.01n)
---> 1000.00 = 0.01n + 0.01n2
---> 100,000 = n + n2
---> n2 + n - 100,000 = 0
Using the quadratic formula with a = 1, b = 2, and c = -100,000 ---> n = 316 (approximately)
+23245 On the first day you save $0.01, on the second day you save $0.02, on the third day you save $0.03, etc.
Thus, on the nth day, you save 0.01·n.
This is an arithmetic sequence because each number is 0.01 greater than the previous number.
The formula for the sum of an arithmetic sequence is: Sum = N(F + L)/2
where N = number of terms, F = value of the first term, L = value of the last term.
a) N = 100, F = 0.01, L = 0.01·N = 1.00 ---> Sum = 100(0.01 + 1.00)/2 ---> Sum = $50.50.
b) N = n, F = 0.01, L = 0.01n, Sum = 500.00 ---> 500.00 = n(0.01 + 0.01n)/2
---> 1000.00 = n(0.01 + 0.01n)
---> 1000.00 = 0.01n + 0.01n2
---> 100,000 = n + n2
---> n2 + n - 100,000 = 0
Using the quadratic formula with a = 1, b = 2, and c = -100,000 ---> n = 316 (approximately)
