a Find a rule to determine the number of pennies are saved on each day.
B
b Start by determining how many pennies you need to save.
A
a $50.50
B
b 316 days
Practice makes perfect
a We start by saving a penny on the first day and we save an additional penny each day after that.
ccc
First Day & Second Day & Third Day
1&2 &3We see that the number of pennies form a sequence and its explicit rule is a_i=i. To find the amount of money saved after 100 days, we will first find the total number of pennies and then multiply it by $0.01.
∑_(i=1)^(100)a_i=∑_(i=1)^(100)i
Since the sum of the first n positive integers is found by the formula n(n+1)2, we can find the total number of pennies using the formula.
∑_(i=1)^(100)i=100( 100+1)/2
Let's evaluate the right-hand side of the equation.
We have 5050 pennies after 100 days. Now, we will multiply this number by 0.01.
5050 * 0.01= 50.50
We will have saved $50.50 after 100 days.
b Let's first find how many pennies make $500.
500/0.01=50 000
We need to find after how many days we will have 50 000 pennies. Let d be the number of days needed. The number of pennies should be 50 000 after d days.
∑_(i=1)^di = 50 000
We can use the formula for the sum of the first n positive integers.
∑_(i=1)^di &= 50 000
⇓ &
d(d+1)/2 & = 50 000
Let's solve for d.
The solutions for this equation are d= - 1 ± sqrt(400 001)2. Let's separate them into the positive and negative cases.
d= - 1 ± sqrt(400 001)2
d_1=- 1 + sqrt(400 001)/2
d_2=- 1 - sqrt(400 001)/2
d_1=- 1/2+sqrt(400 001)/2
d_2=- 1/2-sqrt(400 001)/2
d_1 ≈ 315.73
d_2≈ - 316.73
Using the Quadratic Formula, we found that the solutions of the equation are d_1≈ 315.73 and d_2≈ - 316.73. We will use the positive solution since d represents day. Therefore, after 316 days we will have at least $500.
