Find a formula for the general term
of the geometric sequence three, 15, 75, 375, 1875.
Well, our first term is three. So that bit is easy and I’ve got to
work out what the common ratio. And remember, we’re just gonna do a
division of one term divided by its previous term. And the easiest numbers to work
with here I think are gonna be these two, 𝑎 one is three and 𝑎 two is 15. So the common ratio is 𝑎 two
divided by 𝑎 one which is 15 over three which is five. Now remember, we were told in the
question that this is a geometric sequence. So it didn’t matter which pair of
terms — consecutive terms — that we chose; we would have got the same answer 𝑟
equals five. But just by choosing these first
two terms, the numbers were simpler.
So we know that 𝑎 one, the first
term, is three and the common ratio is five. So we can put that into our
formula. And remember, to work out the value
of any particular term in the sequence, what we do is we take the first term and
we’re gonna keep multiplying it by the common ratio. Now, what we have to do if we are
looking for the fifth term, we’ve only had to multiply that first term by the common
ratio four times in order to get that. So whatever term we’re looking for,
it’s the common ratio to the power of that term minus one. And we just worked out that 𝑎 one
was three and 𝑟 is five. So to work out the value of term 𝑛
in this particular sequence, it’s gonna be three times five to the power of whatever
term that is minus one, 𝑛 minus one.