Question

Kimberly needs $100 to purchase a new bike. She decides to save her money... in the first week, she saves $1.00. In the second week, she saves $2.00. In the third week, she saves $4.00. In the fourth week, she saves $8.00. If Kimberly continues this pattern, in which week would she have saved enough money?

A) Week 8
B) week 6
C) week 9
D week 7

Answer 1

By following the geometric sequence of savings Kimberly adheres to, it is calculated that by Week 7 she will have saved more than $100, thus having enough to purchase the new bike.

Step-by-step explanation:

Kimberly needs $100 to purchase a new bike and saves money in a pattern where she doubles her savings each week ($1 in the first week, $2 in the second, $4 in the third, and so on). To determine in which week she would have saved enough money, we can identify that the savings pattern follows a geometric sequence with a common ratio of 2. The sum of a geometric series Sn can be calculated with the formula Sn = a(1 - r^n) / (1 - r), where 'a' is the first term, 'r' is the common ratio, and 'n' is the number of terms. In this scenario, a = 1, r = 2.

We need to find the smallest 'n' such that Sn >= 100. If we calculate the total sum for 7 weeks we get:

S7 = 1(1 - 2^7) / (1 - 2) = 1(1 - 128) / -1 = 1 * 127 = $127

So, by the end of the seventh week, she would have saved more than $100. Therefore, the correct answer is Week 7 (Option D).