A man has started working with a yearly salary of 14700 Egyptian pounds. Find his salary after six years if he gets a raise of 600 Egyptian pounds every year.
This man is getting the same raise on top of his yearly salary every year. It’s a raise of 600 Egyptian pounds. We could just answer this question by adding 600 to his starting salary of 14700 each year. In year two, his salary will be 15300 Egyptian pounds. In year three, 15900 Egyptian pounds. In year four, 16500 Egyptian pounds. 17100 Egyptian pounds in year five. And finally, 17700 Egyptian pounds in year six. But this isn’t really a smart way to do this question. And if we’d been asked for his salary after 20 years, for example, it would get very tedious going through this process.
Let’s see if there’s a better way. We notice that the yearly salaries paid to this man form the terms of an arithmetic sequence because they have a constant common difference of 600. An arithmetic sequence is just one in which the next term is found by adding or subtracting a single number each time.
We have a formula for working out the general term 𝑇 𝑛 of an arithmetic sequence. We take the first term in our sequence, which is usually called 𝑎, and add to it 𝑛 minus one lots of 𝑑, where 𝑑 represents the common difference. So, for example, to find the second term in the sequence, we take the first term and add two minus one. That’s one lot of the common difference.
To find the third term, we take the first term and add two lots of the common difference. That’s three minus one. To find the sixth term in this sequence then, we take the first term, the man’s salary in the first year, which is 14700. And we add six minus one. That’s five lots of the common difference of 600. Five multiplied by 600 is 3000. And adding this to the starting value of 14700 gives 17700, which is the same as the value we found when we went through the process working out the salary for each year individually.
The second method is the smart method. And it’s certainly going to be much quicker as the number of years increases.
We found that this man’s salary after six years is 17700 Egyptian pounds.