Write the next three terms of the following sequence: one, 16, 81, blank, blank, blank.
So as we begin this question on sequences, let’s remind ourselves that the word terms here refers to the numbers in the sequence. So here, we’re given the first three terms, term one, two, and three. And we need to find term four, five, and six. So let’s start by looking at the terms and seeing if we can find a common difference between them. If we take a look at our first and second terms, we can find the difference is 15 by working out 16 take away one. And from 16 to 81, there’s a difference of 65. In this sequence, there’s not a common difference. We don’t have the same number, 15 and 15 or 65 and 65.
There’s also not an obvious pattern or link between 15 and 65. For instance, 65 is not a multiple of 15. So since we don’t have a pattern between the terms of each number in the sequence, let’s go back and see if we can find a link between the term number and the term itself. Let’s see if we could find a link between term number one and the value one, term number two and the value 16, and term number three and the value 81.
So let’s create a table with our term numbers — one, two, three, and so on — written and the terms of our sequence close by so that we can compare the values. In a sequence like the one we have here, where there is no obvious pattern between the terms, then it’s very likely that we’re going to have exponents. And by exponents, that just means we’re going to look at squares or we’re taking the third, fourth, or fifth powers of the number. So let’s have a look at our first option.
Here, we’re going to try squaring the term number. That means we’re going to work out one squared, two squared, and three squared. And since squaring means multiplying by itself, for our first term number, we’re going to have one squared which is one times one, giving us one. In our second column, we have two squared which is two times two, giving us the value four. And in our next column then, three squared is three times three, which is nine. So now, we need to check if the values we find by squaring the term number are the same as the actual terms of our sequence. That is, our one, four, and nine the same as one, 16, and 81. No, they’re not. So let’s try a second option.
Here, we’re going to find the third power of each term number. That means we’re going to work out one to the third power, two to the third power, and three to the third power. And since taking a number to the third power means multiplying it by itself three times, when we calculate one cubed, we’re going to be working out one times one times one which gives us one. And for two cubed, we’re going to be calculating two times two times two. And when we’re calculating this, we’ve got to be a little bit careful. If we work out two times two, that will give us four. And then we multiply the answer four by two, giving us eight.
And in our next column, we have three to the third power or three cubed, which is three times three times three. And we can evaluate this by saying our first three times three will give us nine. And then multiply it by another three will give us 27. So now we go back and check. Does term number to the third power gave us the actual terms of the sequence? That is, is one, eight, and 27 the same as one, 16, and 81? Unfortunately, no, it doesn’t. So let’s try another exponent of term number.
Let’s try finding the fourth power of our term numbers. So we’re going to calculate one to the fourth power, two to the fourth power, and three to the fourth power. So since taking the fourth power of a number means multiplying it by itself four times. That means for one to the fourth power, we’re going to calculate one times one times one times one, giving us the answer one. Next, we have two to the fourth power, which is two times two times two times two, which we can calculate in stages. We know that two times two will give us four. Then, if you multiply four by two, that will give us eight. And then multiplying by the final two will give us 16. So two to the fourth power is 16.
Next, we have three to the fourth power, which is three times three times three times three, which we can also work out in stages. If we work out just three times three, that will give us nine. And then multiplying nine by the next three will give us 27. And then working out 27 times our final three will give us 81. So three to the fourth power is 81. And so, now we need to check if our term numbers to the fourth power will give us the actual terms of our sequence. And yes, in this case, they do.
So now, the question asked us for the next three terms of the sequence. So we can use the rule “term number to the fourth power” to help us work these out. And we can do this by taking the term numbers four, five, and six and taking those to the fourth power. We don’t need to worry about doing anything with the term number squared or to the third power because they weren’t useful to us.
Let’s start by working on our term number four to the fourth power. This means that we need to calculate four times four times four times four. So the first four times four is 16. And then, we can multiply 16 by the next four, giving us 64. And then, finally, we calculate 64 times the final four, which gives us 256. And in our next column, we’re going to work out five to the fourth power, which is five times five times five times five, which will give us an answer of 625. And for our final calculation then, we work out six to the fourth power, which will give us 1296.
So in summary then, in this question, we were given the first three terms. We used these terms to help us work out a rule for the sequence. And we used this rule to help us find the next three terms. So answer for these three terms is 256, 625, and 1296.