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?

Step-by-step explanation:

The formula for determining the sum of n terms of an arithmetic sequence is expressed as

Sn = n/2[2a + (n - 1) d]

Where

n represents the number of terms in the arithmetic sequence.

d represents the common difference of the terms in the arithmetic sequence.

a represents the first term of the arithmetic sequence.

From the information given,

a = 1 penny = 1/100 = $0.01

d = 0.01

a) For 100 days, the sum of the first 100 terms, S100 would be

S100 = 100/2[2 * 0.01 + (100 - 1) 0.01]

S100 = 50[0.02 + 0.99)

S100 = 50 * 1.01 = $50.5

b) when Sn = $500, then

500 = n/2[2 * 0.01 + (n - 1) 0.01]

Multiplying through by 2, it becomes

500 * 2 = n[2 * 0.01 + (n - 1) 0.01]

1000 = n[0.02 + 0.01n - 0.01]

1000 = n[0.01 + 0.01n]

1000 = 0.01n + 0.01n²

0.01n² + 0.01n - 1000 = 0

Applying the general formula for quadratic equations,

x = [-b±√ (b² - 4ac) ]/2a

n = - 0.01±√0.01²-4 (0.01 * - 1000) ]/2 * 0.01

n = ( - 0.01 ± √40.001) / 0.02

n = ( - 0.01 + 6.32) / 0.02 or

n = ( - 0.01 - 6.32) / 0.02

n = 315.5 or n = - 316.5

Since n cannot be negative, then n = 315.5

It will take approximately 316 days to save $500