Answer:
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