Question Video: Finding the Formula of the General Term of a Geometric Sequence Mathematics

Find a formula for the general term of the geometric sequence three, 15, 75, 375, 1875, โ€ฆ.

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Video Transcript

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 Iโ€™ve got to work out what the common ratio. And remember, weโ€™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, ๐‘Ž one is three and ๐‘Ž two is 15. So the common ratio is ๐‘Ž two divided by ๐‘Ž 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 didnโ€™t matter which pair of terms โ€” consecutive terms โ€” that we chose; we would have got the same answer ๐‘Ÿ equals five. But just by choosing these first two terms, the numbers were simpler.

So we know that ๐‘Ž 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 weโ€™re gonna keep multiplying it by the common ratio. Now, what we have to do if we are looking for the fifth term, weโ€™ve only had to multiply that first term by the common ratio four times in order to get that. So whatever term weโ€™re looking for, itโ€™s the common ratio to the power of that term minus one. And we just worked out that ๐‘Ž one was three and ๐‘Ÿ is five. So to work out the value of term ๐‘› in this particular sequence, itโ€™s gonna be three times five to the power of whatever term that is minus one, ๐‘› minus one.

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