Video Transcript
Benjamin saves one pound on the first day, two pound on the second day, three pounds on the third day, and so on, saving an extra one pound each day. On which day will he have saved over 100 pounds in total?
Let’s look at the sequence formed by the amounts Benjamin saves each day. On the first day, it’s one pound; on the second day, it’s two pound; on the third day, it’s three pound; and so on. The amounts Benjamin saves increase by one pounds each day. The difference between the terms in this sequence is therefore constant. And so the terms form an arithmetic sequence with a common difference of one. We are asked to find the day on which he will have saved over 100 pounds in total. That means we’re looking for the day on which the sum of the terms first exceeds 100 pounds.
We recall the formula for finding the sum of the first 𝑛 terms of an arithmetic sequence. It’s 𝑆 sub 𝑛 equals 𝑛 over two multiplied by two 𝑎 plus 𝑛 minus one 𝑑, where 𝑆 sub 𝑛 represents the sum of the first 𝑛 terms. 𝑛 represents the number of terms whose sum we’re finding. 𝑎 or sometimes 𝑎 sub one represents the first term in the sequence. And 𝑑 represents the common difference. The first term and the common difference for this sequence are each one. The sum of the first 𝑛 terms we want to be 100 and 𝑛, the number of terms, we don’t know. That’s what we’re trying to work out. We can therefore form an equation by substituting 100 for 𝑆 sub 𝑛, one for 𝑎, and one for 𝑑. We have 100 equals 𝑛 over two multiplied by two times one plus 𝑛 minus one multiplied by one.
This equation simplifies to 100 equals 𝑛 over two multiplied by two plus 𝑛 minus one. And within the parentheses, this simplifies further to 𝑛 plus one. We can then multiply both sides of the equation by two to give 200 equals 𝑛 multiplied by 𝑛 plus one. Distributing the parentheses, on the right-hand side, we have 200 is equal to 𝑛 squared plus 𝑛. And finally, we can subtract 200 from each side of the equation to give zero is equal to 𝑛 squared plus 𝑛 minus 200. We now have a quadratic equation in 𝑛. This equation can’t be factored, so we’ll solve by applying the quadratic formula. Remember, the quadratic formula tells us that the roots of the equation 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 is equal to zero are given by 𝑥 equals negative 𝑏 plus or minus the square root of 𝑏 squared minus four 𝑎𝑐 all over two 𝑎.
The coefficients in our quadratic equation are one, one, and negative 200. So we have 𝑛 is equal to negative one plus or minus the square root of one squared minus four multiplied by one multiplied by negative 200 all over two multiplied by one. This simplifies to 𝑛 equals negative one plus or minus the square root of 801 all over two. If we then evaluate on our calculators, we find that 𝑛 is equal to 13.650 or negative 14.650. Remember though that 𝑛 represents a number of terms, and so it must be positive. We can therefore eliminate the solution of negative 14.650. 𝑛 must also be an integer. So let’s think what this value of 13.650 tells us.
If 𝑆 sub 𝑛 were a continuous function of 𝑛, then it would be equal to 100 when 𝑛 equals 13.650, which means that on day 13, 𝑆 sub 𝑛 would be less than 100, whereas on the next day, day 14, 𝑆 sub 𝑛 would be greater than 100. Remember we’re looking for on which day Benjamin will have saved over 100 pounds in total. On day 13, he hasn’t quite got there, but on day 14, he’s gone past 100 pounds. So it’s day 14. That’s our answer. We can check this by evaluating 𝑆 sub 13 and 𝑆 sub 14. 𝑆 sub 13 is equal to 13 over two multiplied by two plus 12 times one. That’s two times one for two 𝑎 plus 𝑛 minus one 12 multiplied by the common difference 𝑑, which is one. This evaluates to 91. So the total amount Benjamin will have saved on day 13 is 91 pounds.
On day 14, 𝑆 sub 14 is equal to 14 over two multiplied by two plus 13 times one, which is 105. This confirms that Benjamin will have saved less than 100 pounds in total on the 13th day, but more than 100 pounds in total on the 14th day. So this confirms that our answer of 14 is correct for the day on which Benjamin will have saved over 100 pounds in total.
