Question

1. isabella saved 2 nickels today. if she doubles the number of nickels she saves each day, how many days, including today, will it take her to save more than 500 nickels?

Explanation:

Step1: Identify the sequence type

This is a geometric sequence where the first term \(a_1 = 2\) and the common ratio \(r = 2\) (since she doubles the number of nickels each day). The sum of a geometric sequence is given by the formula \(S_n=\frac{a_1(1 - r^n)}{1 - r}\) (for \(r
eq1\)). We need to find the smallest \(n\) such that \(S_n>500\).

Step2: Substitute the values into the formula

Substitute \(a_1 = 2\), \(r = 2\) into the sum formula: \(S_n=\frac{2(1 - 2^n)}{1 - 2}=\frac{2(1 - 2^n)}{-1}=2(2^n - 1)=2^{n + 1}-2\).

Step3: Solve the inequality \(2^{n+1}-2>500\)

First, add 2 to both sides: \(2^{n + 1}>502\).
Now, we can test values of \(n\):

• For \(n = 8\): \(2^{8 + 1}=2^9 = 512\). Then \(2^{9}-2=512 - 2 = 510>500\).
• For \(n = 7\): \(2^{7+1}=2^8 = 256\). Then \(2^{8}-2=256 - 2 = 254<500\).

Answer:

8