: Enter your answer and show all the steps that you use to solve this problem in the space provided.
The deposits Ginny makes at her bank each month form an arithmetic sequence. The deposit for month 3 is $150, and the deposit for month 5 is $180. Answer the questions below and show all work.
What is the common difference for the deposits made each month?
Write an explicit formula for this arithmetic sequence.
What is the amount of Ginny’s deposit in the 12th month?
At what month will Ginny first make a deposit that is at least $500?
(4 points)
To find the common difference for the deposits made each month, we can use the given information for month 3 and month 5:
Month 3: $150
Month 5: $180
Using the formula for the nth term of an arithmetic sequence:
An = A1 + (n-1)d
Substitute the values for month 3 and 5:
$150 = A1 + 2d
$180 = A1 + 4d
Subtract the first equation from the second equation:
$180 - $150 = A1 + 4d - A1 - 2d
$30 = 2d
d = $15
The common difference for the deposits made each month is $15.
Next, we can write an explicit formula for this arithmetic sequence:
An = A1 + (n-1)d
An = A1 + (n-1)*15
An = A1 + 15n - 15
To find the amount of Ginny's deposit in the 12th month, substitute n = 12:
A12 = A1 + 15*12 - 15
A12 = A1 + 180 - 15
A12 = A1 + 165
To find A1, we can use the information for month 3:
$150 = A1 + 2*15
$150 = A1 + 30
A1 = $120
Now, substitute A1 = $120 into the equation for the 12th month:
A12 = $120 + 165
A12 = $285
Therefore, Ginny's deposit in the 12th month is $285.
To find at what month Ginny will first make a deposit that is at least $500, we can set up the inequality:
A1 + 15n - 15 ≥ $500
Substitute A1 = $120:
$120 + 15n - 15 ≥ $500
15n ≥ $395
n ≥ 26.33
Since n must be a whole number, Ginny will first make a deposit that is at least $500 in the 27th month.