Video Transcript
Find the first five terms in the
sequence 𝑎 𝑛 is equal to two times 𝑎 𝑛 minus one plus 𝑎 𝑛 minus two, where 𝑎
one is three, and 𝑎 two is five, and 𝑛 is greater than or equal to three, and 𝑛
is an integer.
Now, this formula is telling us
that a term is some combination of the previous two terms. It’s twice the immediately
preceding term, and then we have to add on the term before that. And to get this sort of formula
going, we need to provide two initial terms, the first and the second term, to
enable you to work out the third term. And now, 𝑛 is greater than or
equal to three. If 𝑛 was less than three, if 𝑛
started at two, then we’d have the second term is equal to the first term- two times
the first term plus the zeroth term. So, that wouldn’t work. So, this recursive sequence is only
set up for 𝑛 is greater than or equal to three.
Well, the question asked us to work
out the first five terms. We’ve already been given the first
two, so let’s have a go at working out the third one. And as we said in the formula, to
work out a particular term, we double the previous term, 𝑛 minus one, and then we
add on the term before that, 𝑛 minus two. Let’s just plug in those values for
𝑎 one and 𝑎 two then. So, 𝑎 two is five and 𝑎 one is
three. So, 𝑎 three is two times five plus
three, which is 10 plus three, and that’s 13.
Now, the fourth term is twice the
third term plus the second term. And that’s two times 13 plus
five. We’ve just worked out the third
term. Well, two times 13 is 26. So, 26 plus five is 31. And then, finally, the fifth term
is two times the fourth term plus the third term. And plugging in the values of the
fourth and third term that we’ve just worked out, we can see that the fifth term is
75. And we just need to write all those
out in full for our answer.
