Cassandra starts working for a company for an annual salary of $30,000. If she earns a 2% raise at the end of each year, what will her total earnings be after five years. I have to determine the type of series----identify the first term--the common difference or the common ratio, and the number of terms--the general formula to use to substitute and simplify.
Cassandra starts working for a company for an annual salary of $30,000. She is given a 2% raise at the end of each year.
Her earnings after n year of working is given by `30000*(1+2/100)^n`
`30000*(1+2/100)^n` is a term of a geometric series with first term 30000 and common ratio 1.02.
The sum of n terms of a geometric series is `(A*(r^n - 1))/(r - 1)` .
Here, A = 30000, r = 1.02 and n = 5
The required sum is `30000*(1.02^5 - 1)/(1.02 - 1)`
`A = P(1+r/n)^(t*n)`
$30,000 is the initial amount (P)
1 + r stands for the intial amount plus the the rate, and much it is compounded.
t*n stands for the number of years times the number of payments
If she earns a 2% raise, that means she the rate is .02%
`A = 30,000 (1+.02/1)^(5*1)`
`A = 30,000(1.02)^5`
`A = $33,122.42` this is the amount of money after 5 years in the bank
The type of series is geometric series because the common formula for geometric is `A_n = A_1 + r^(n-1)`
The first term in the series is 30,000, and the ratio is 1.2. It is not .2, but 1.2, because you are increasing, and it is the overall total.
To find what their salary would be after 5 years, starting at 30000 per year, with a 2% raise each year, you would use the formula 30000 (1 + 2/100) ^ n.
Your first term would be 30000 and the common ratio is (1 + 2/100) or 1.02. N would be 5, for 5 years.
I am not really sure how to solve geometric series since it has been a while but I put a couple links that might help you :)