Suppose you sign a contract for an annual salary of $50,000 with a guaranteed raise of 5% each year.
1.Write your salary for the next n years as a geometric sequence in explicit form.
2.What will your salary be in year 5?
3.How much will you have earned in total salary by the end of your 10th year?
I got 2.60,000 3.70,000
an = a r^(n-1)
= 50,000 (1.05)^(n-1)
a5 = 50,000 (1.05)^4 = 60,775.31
sum 1 to 10
= a (1-r^n)/(1-r)
= 50,000 (1-1.05^10)/(-.05)
= 50,000 ( -.6289 / -.05)
= 628,894.63
To answer these questions, we need to use the formula for a geometric sequence. A geometric sequence is a sequence of numbers in which each term after the first is found by multiplying the previous term by a common ratio.
1. Write your salary for the next n years as a geometric sequence in explicit form:
In this case, the initial term is $50,000, and the common ratio is 1 + 5% = 1.05.
The explicit formula for a geometric sequence is given by:
an = a1 * r^(n-1)
In this formula,
an represents the nth term in the sequence,
a1 represents the first term in the sequence,
r represents the common ratio, and
n represents the position of the term in the sequence.
So, for this scenario, the explicit form of the geometric sequence representing your salary for the next n years is:
salary(n) = $50,000 * (1.05)^(n-1)
2. To find the salary in year 5, we substitute n = 5 into the formula:
salary(5) = $50,000 * (1.05)^(5-1)
= $50,000 * (1.05)^4
= $50,000 * 1.2155
= $60,775
Therefore, your salary in year 5 would be $60,775.
3. To find the total earnings by the end of your 10th year, we need to sum up the salaries from year 1 to year 10. We can use the formula for the sum of a geometric sequence:
Sn = a1 * (r^n - 1) / (r - 1)
In this formula,
Sn represents the sum of the first n terms in the sequence.
So, for this scenario, the total salary by the end of your 10th year is:
salary(10) = $50,000 * (1.05^10 - 1) / (1.05 - 1)
= $50,000 * (1.629 - 1) / 0.05
= $50,000 * 0.629 / 0.05
= $629,000 / 0.05
= $12,580,000
Therefore, you will have earned a total salary of $12,580,000 by the end of your 10th year.
It looks like the values you shared, 2.60,000 for the salary in year 5 and 3.70,000 for the total earnings by the end of the 10th year, are incorrect. The correct values are $60,775 for the salary in year 5 and $12,580,000 for the total salary by the end of the 10th year.
To answer the questions, let's break it down step by step:
1. Write your salary for the next n years as a geometric sequence in explicit form.
A geometric sequence is a sequence where each term is found by multiplying the previous term by a constant. In this case, your initial salary is $50,000, and you have a guaranteed raise of 5% each year. So, your salary for each year can be written as:
a(n) = a(1) * r^(n - 1)
Where a(n) represents the salary in year n, a(1) represents the initial salary ($50,000), r represents the rate of increase (1 + 5% = 1.05), and n represents the number of years.
2. What will your salary be in year 5?
To find the salary in year 5, we will substitute n = 5 into the geometric sequence formula:
a(5) = a(1) * r^(5 - 1)
= $50,000 * (1.05)^(4)
≈ $61,578.13
So, your salary in year 5 will be approximately $61,578.13.
3. How much will you have earned in total salary by the end of your 10th year?
To find the total salary earned by the end of the 10th year, we will sum up the salaries for each year from year 1 to year 10. The formula to calculate the sum of a geometric sequence is:
S(n) = a(1) * (r^n - 1) / (r - 1)
Where S(n) represents the sum of the salaries up to year n. Let's calculate the total salary for the 10th year:
S(10) = a(1) * (r^10 - 1) / (r - 1)
= $50,000 * (1.05^10 - 1) / (1.05 - 1)
≈ $638,147.95
So, you will have earned approximately $638,147.95 in total salary by the end of your 10th year.