Homework Help: Quick Math Help
Posted by Anonymous on Tuesday, October 11, 2016 at 4:06pm.
1. let an=4n find the sum of the first 5 terms.
How would you do this?
2. a1=3; an+1=3an+18. Find a6
How would you do this?
3. You start you 30 year career at a salary of $50,000 and receive yearly raises of $1,000. How much money will you have earned, in total, over the 30 years?
1. list the terms
a(n) = 4n, start with n=1,2,..
terms are 4, 8,12, 16, 32
now add them up
2. I think you mean
a(n+1) = 4a(n) + 18, where a(1) = 3
a(2) = 4a(1) + 18 = 4(3)+18 = 30
a(3) = 4a(2) + 18 = 4(30)+18 = 138
carry on to a(6)
This is called a recursive sequence, where any term depends on the value of the previous term
3. An AS, where a = 50000, and d = 1000
n = 30
Use your formula for the sum of n terms
1. To find the sum of the first 5 terms in the series with the formula an = 4n, you can use the formula for the sum of an arithmetic series.
The formula for the sum of an arithmetic series is Sn = (n/2)(a1 + an), where Sn is the sum of the first n terms, a1 is the first term, and an is the nth term.
In this case, the first term, a1, is 4, and we want to find the sum of the first 5 terms, so n = 5.
Plugging in these values into the formula, we get:
S5 = (5/2)(4 + a5)
To find a5, we substitute n = 5 into the given formula for an:
a5 = 4 * 5 = 20
Now we can plug this value back into the equation for the sum:
S5 = (5/2)(4 + 20)
S5 = (5/2)(24)
S5 = 60
Therefore, the sum of the first 5 terms in the series an = 4n is 60.
2. To find a6 in the sequence given by a1 = 3 and an+1 = 3an + 18, we can use the recursive formula to generate each term in the sequence.
Starting with a1 = 3, we can find a2 using the recursive formula:
a2 = 3a1 + 18 = 3 * 3 + 18 = 27
Similarly, we can find a3, a4, a5, and finally a6 by repeatedly applying the recursive formula.
a3 = 3a2 + 18 = 3 * 27 + 18 = 99
a4 = 3a3 + 18 = 3 * 99 + 18 = 315
a5 = 3a4 + 18 = 3 * 315 + 18 = 963
a6 = 3a5 + 18 = 3 * 963 + 18 = 2895
Therefore, a6 is equal to 2895 in the given sequence.
3. To calculate the total amount of money earned over a 30-year career with a starting salary of $50,000 and yearly raises of $1,000, you can use the formula for the sum of an arithmetic series.
In this case, the first term, a1, is $50,000, and the common difference, d, is $1,000. We want to find the sum of the first 30 terms, so n = 30.
Using the formula for the sum of an arithmetic series, Sn = (n/2)(a1 + an), we can calculate the total amount earned over 30 years.
Plugging in the values, we get:
S30 = (30/2)($50,000 + a30)
To find a30, we can use the formula for the nth term in an arithmetic series:
an = a1 + (n-1)d
Plugging in the values, we get:
a30 = $50,000 + (30-1)($1,000)
a30 = $50,000 + 29($1,000) = $50,000 + $29,000 = $79,000
Now we can substitute this value back into the equation for the sum:
S30 = (30/2)($50,000 + $79,000)
S30 = 15($129,000) = $1,935,000
Therefore, over the 30-year career, you will have earned a total of $1,935,000.