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.
