Question Video: Finding the First Five Terms of a Sequence Given Its General Term Mathematics

Video Transcript

Find the first five terms of the sequence whose 𝑛th term is given by 𝑎 sub 𝑛 equals five 𝑛 squared plus 𝑛 cubed.

Let’s begin this question by just recalling a little bit about what we mean by sequences and the 𝑛th term or general term. The letter 𝑛 represents the index of the sequence. The index value will always be an integer. And unless we’re told otherwise, we assume that this index starts with a value of one. So, we would say that the first term in the sequence can be written as 𝑎 sub one. The second term has an index 𝑛 equal to two and can be written as 𝑎 sub two. We can continue this pattern. And this allows us to say that any term with an index 𝑛 can be represented as 𝑎 sub 𝑛.

And as we are given here, knowing this equation for the 𝑛th term of a sequence allows us to generate any term in the sequence. We simply substitute the index number of the term that we wish to find out into the 𝑛th term. For example, if we wanted to find the 10th term, we would substitute 𝑛 is equal to 10. Because we want to find the first five terms of this sequence, we’re going to substitute 𝑛 equals one, two, three, four, and five into the 𝑛th term.

Let’s start by substituting in 𝑛 is equal to one. This gives us 𝑎 sub one is equal to five times one squared plus one cubed. And we must be really careful to apply the order of operations, particularly in the first part of this expression on the right-hand side. When we have five 𝑛 squared, that means that we square the 𝑛 first and then multiply by five. So, five times one squared is equal to five times one plus one will give us a six. That means that the first term in this sequence is six.

To find the second term, we substitute 𝑛 equals two, which gives us 𝑎 sub two is equal to five times two squared plus two cubed. Remember that we square the two first, that’s four, multiply it by five gives us 20, and then two cubed is equal to eight. And that gives us that the second term is equal to 28. Then, 𝑎 sub three is equal to five times three squared plus three cubed. Simplifying this, we get a third term of 72. In the same way, by substituting 𝑛 is equal to four, we find that the fourth term is 144. By substituting 𝑛 is equal to five, we find that 𝑎 sub five is equal to 250.

We can therefore give the answer that the first five terms in the sequence are six, 28, 72, 144, and 250.