Salary If you take a job with a starting salary of 35,000...

Salary If you take a job with a starting salary of 35,000 dollar per year and a guaranteed raise of 1400 dollar per year, how many years will it be before your aggregate salary is 280,000 dollar ? [Hint: Remember that your aggregate salary after 2 years is $35,000+( 35,000+ 1400) \cdot]$

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a series of numbers in which each term after the first is obtained by adding a constant difference to the preceding term. This concept is used when dealing with problems involving steady increases or decreases, such as a fixed annual raise in a salary.

Arithmetic Series

An arithmetic series is the sum of all the terms in an arithmetic sequence. The series allows one to calculate the aggregate total over a period by using a formula that incorporates the first term and the common difference, which is essential when computing cumulative salaries over several years.

Solving Linear Equations

Many aggregate problems eventually lead to a linear equation that must be solved to find the unknown variable, such as the number of years needed to reach a specified total. This process involves setting up an equation based on the arithmetic series formula and isolating the variable to determine the solution.

Transcript

00:01 So in this problem, we're told that you're going to take a job and your starting salary is going to be $35 ,000 per year.

00:07 And then you're going to get a guaranteed raise of $1 ,400 per year.

00:11 So if your salary is going to keep increasing by $400 per year, this is kind of like an arithmetic sequence, where a sub 1 would represent $35 ,000.

00:20 And d, our common difference would be $1 ,400.

00:23 So what we're trying to figure out is how many years is it going to be before your aggregate salary is $280 ,000? basically that's the sum of all of your salaries.

00:33 So what we're trying to figure out is, well, how long is it going to take for that? so that means our sum, s of n, would be $280 ,000.

00:42 So in order to do this, we're going to use one of our sum formulas.

00:46 So i'm going to use the sum formula where s of n is equal to n divided by two times the quantity of two times a sub one plus d times m minus one.

00:58 So let's substitute in what we know.

00:59 We know s of n is equal to 280 ,000.

01:03 We don't know n.

01:04 That's what we're looking for.

01:06 A sub 1 is 35 ,000.

01:08 So we have 2 times 35 ,000 plus d, which is 1 ,400, which is getting multiplied by n minus 1.

01:16 So now we just need to solve this equation for n.

01:19 So first, i'm going to bring down the 280 ,000.

01:23 We're going to leave n over 2 for now.

01:25 2 times 35 ,000 is 70 ,000.

01:28 And when we distribute the 1400, we get 1400n minus 1400.

01:36 Well, i see we have late terms.