Question Video: Finding the Terms of a Sequence given Its General Term and the Value of the First Term Mathematics • 9th Grade

Find the first five terms of the sequence with the general term 𝑎_(𝑛 + 1) = 5𝑎_𝑛, where 𝑛 ≥ 1 and 𝑎_(1) = 2.

02:06

Video Transcript

Find the first five terms of the sequence with the general term 𝑎 sub 𝑛 plus one is equal to five times 𝑎 sub 𝑛, where 𝑛 is greater than or equal to one and 𝑎 sub one equals two.

Let’s think about what this general term is saying. The term 𝑛 plus one will be equal to the previous term, term 𝑛, multiplied by five. We already know our first term equals two. Our second term, 𝑎 sub two, would be the term of 𝑎 sub one plus one, which means 𝑎 sub two will be equal to five times 𝑎 sub one. The second term is equal to five times the first term. Since our first term was two, our second term will be 10.

Now we should be able to see a pattern. The third term will be equal to five times the second term, which makes the third term 50. It’s worth noting here that we’re dealing with a geometric sequence. In a geometric sequence, each term is found by multiplying the previous term by a constant called the common ratio. Since 𝑎 sub 𝑛 plus one is equal to five times 𝑎 sub 𝑛, this is a geometric sequence with a common ratio of five.

So we started with two and multiplied that by five to get the second term of 10, which we multiplied by five to get the third term 50. To find the fourth term, we multiply 50 by five, and we get 250. And to find our fifth term, we multiply 250 by five, which gives us 1,250.

Listing the first five terms of this sequence, we get two, 10, 50, 250, 1,250.