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 these questions
1.what is common differences for each month? $15
2.write explicit formula for arithmetic sequence
3.amount in 12th month?
4.what month will deposit at least $500?
3 150
4 150 + d
5 150 + 2 d = 180
so d = 30/2 = 15
month 1 = 150-30 = 120
An = 120 + (n-1)15
A12 = 120 + 11(15) = 285 yes correct
500 = 120 + (n-1)15
n - 1 = 25.33
n = 26.33
yes, 27
105+15n
285
month 27
these are my other answers
1. The common difference for each month is $15.
2. To write the explicit formula for an arithmetic sequence, we can use the formula: aₙ = a₁ + (n-1)d, where aₙ represents the nth term, a₁ represents the first term, n represents the term number, and d represents the common difference.
In this case, the first term (a₁) is $150 and the common difference (d) is $15. Therefore, the explicit formula for this arithmetic sequence is:
aₙ = 150 + (n-1)15
3. To find the amount in the 12th month, we substitute n = 12 into the explicit formula:
a₁₂ = 150 + (12-1)15
a₁₂ = 150 + 11 * 15
a₁₂ = 150 + 165
a₁₂ = $315
Therefore, the amount in the 12th month is $315.
4. To find the month when the deposit is at least $500, we can set up an inequality using the explicit formula:
aₙ ≥ 500
150 + (n-1)15 ≥ 500
Simplifying the inequality, we get:
(n-1)15 ≥ 350
Dividing both sides by 15:
n-1 ≥ 23.33
Since n represents the month number, it can only be a whole number. Thus, we take the smallest whole number greater than or equal to 23.33, which is 24.
Therefore, the deposit will be at least $500 starting from the 24th month.
To answer these questions, we need to understand arithmetic sequences.
An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. Let's use the given information to find the common difference and the explicit formula for the sequence.
1. Common Difference:
The deposit for month 3 is $150, and the deposit for month 5 is $180. To find the common difference, we can subtract the deposit for month 3 from the deposit for month 5:
Common difference = deposit for month 5 - deposit for month 3 = $180 - $150 = $30.
Therefore, the common difference for each month is $30.
2. Explicit Formula:
The explicit formula for an arithmetic sequence is given by the formula:
an = a1 + (n-1)d
In this formula, "an" represents the nth term in the sequence, "a1" is the first term of the sequence, "n" is the term number, and "d" is the common difference.
Using the given information, we can write the explicit formula:
an = a1 + (n-1)d
an = 150 + (n-1)30
an = 150 + 30n - 30
an = 120 + 30n
So, the explicit formula for the arithmetic sequence is an = 120 + 30n.
3. Amount in the 12th Month:
To find the amount in the 12th month, we substitute n = 12 into the explicit formula:
a12 = 120 + 30 * 12
a12 = 120 + 360
a12 = 480.
Therefore, the amount in the 12th month is $480.
4. Month to Deposit at Least $500:
To find the month in which the deposit will be at least $500, we need to solve the equation an ≥ 500 with the given explicit formula:
120 + 30n ≥ 500
To isolate "n," we subtract 120 from both sides:
30n ≥ 380
Finally, divide both sides by 30:
n ≥ 12.67
Since "n" represents the number of months, we round up to the nearest whole number because you can't have a fraction of a month. Therefore, the deposit will be at least $500 in the 13th month.
In summary:
1. The common difference for each month is $30.
2. The explicit formula for the arithmetic sequence is an = 120 + 30n.
3. The amount in the 12th month is $480.
4. The deposit will be at least $500 in the 13th month.