### Video Transcript

A company wants to distribute 14,500 Egyptian pounds among the top five sales representatives as a bonus. The bonus for the last-placed representative is 1,300 Egyptian pounds, and the difference in bonus is constant among the representatives. Find the bonus of the representative in the first place.

We have five sales representatives ordered from first to fifth who are going to share this money. The bonus for the last-placed representative, that’s the representative in fifth place, we’re told is 1,300 Egyptian pounds. We’re also told that the difference in the bonus being paid is constant between the representatives, which means that these amounts form an arithmetic sequence with a common difference of 𝑑.

We don’t know what this common difference is, but as the bonuses are decreasing, we know that its value will be negative. We’ve also been given the total amount of money that is going to be shared between these five people. This is the sum of the five terms in our arithmetic sequence. We know that there is a formula for calculating the sum of the first 𝑛 terms in an arithmetic sequence. It’s 𝑆 sub 𝑛 is equal to 𝑛 over two multiplied by two 𝑎 plus 𝑛 minus one 𝑑, where 𝑎 represents the first term in the sequence and 𝑑 represents the common difference.

We can therefore form an equation in 𝑎 and 𝑑. Substituting 14,500 for the sum of the five terms and five for 𝑛, we have 14,500 is equal to five over two multiplied by two 𝑎 plus four 𝑑. We can simplify this equation slightly by first canceling a factor of two on the right-hand side to give 14,500 equals five over one, or five, multiplied by 𝑎 plus two 𝑑. And then we can divide both sides of the equation by five to give 2,900 is equal to 𝑎 plus two 𝑑.

Now we have an equation connecting 𝑎 and 𝑑. But this isn’t enough information to enable us to work out 𝑎 and 𝑑, as we have only one equation and two unknowns. The other information we were given is that the fifth term of the sequence is equal to 1,300. This tells us that if we take our first term 𝑎 and we add the common difference of 𝑑 four times, then we get 1,300. So we have the equation 𝑎 plus four 𝑑 equals 1,300. We now have a pair of linear equations in 𝑎 and 𝑑 which we can solve simultaneously. Subtracting the first equation from the second will eliminate the 𝑎 terms and leave two 𝑑 is equal to negative 1,600.

We can then divide both sides of this equation by two to find that 𝑑 is equal to negative 800. We have a negative value for 𝑑, as we expected, so that’s reassuring. We can then substitute this value of 𝑑 into either of our two equations, I’ve chosen equation one, to give an equation in 𝑎 only. We have 𝑎 minus 1,600 is equal to 2,900. Adding 1,600 to each side of this equation, we find that 𝑎 is equal to 4,500. So we’ve found the bonus paid to the representative in first place.

To check our answer, let’s calculate the bonuses paid to the remaining representatives by subtracting 800 each time. This gives 3,700, 2,900, 2,100. And if we subtract 800 again, this does indeed give 1,300. If we then sum these five values together, it does indeed give 14,500, which is the correct total. So we can confirm that the bonus paid to the representative in first place is 4,500 Egyptian pounds.