Geometric sequences

A geometric sequence is a sequence where every number is separated by the same certain ratio. The ratio is therefore the same for every number in a sequence. In the example

{3, 9, 27, 81, 243}

where the first term is 3 and the second is 9. The ratio here is 3, as 9 devided by 3 is 3. Continuing, we have 27, 81 and 243, which all follow the ratio of 3.

And of course, the it can also go the other way. When the value decreases, the ratio is under 0. We can apply this to the same example used eariler, and get: 

{243, 81, 27, 9, 3}

As you see here, the ratio is not 3, but rather 1/3. If you mutiply 243 by 1/3, you get 81, and so forth.

If presented if this sequence:

{3, 9, 27, 83, 243}

You can probably figure out that the ratio is three, but if you are asked to find the 7th term or 50th term, you will start to run into problems just trying to multiply each number by 3. So as for most things in matheatics, we have formula. The formula for geometric sequences is:

un = u1 × rn-1

where un is the term you want to find, u1 is the first term of the sequence, r is the ratio in the sequence and n is the term number. If you want to find the 7th term in the sequence stated above, you would do this:

un = u1 × rn-1

u7 = 3 × 37-1

u7 = 3 × 729

u7 = 2187

So the 7th term is 2187. The same method goes for an other term.

When presented with a geometric sequence, you are often asked to find something else than just the term. Sometimes you might need to find multiple things to find the aswer. For example, given r is 4 and the 5th term is 4352 what is u1?

In this example we are given  the ratio ,4, the term number,5, and the value of term 5, 4352. If we plug those values into the general formula, we get

4352 = u1 × 45-1

which means

4352 = u1 × 44

divide both sides by 44 

4352/44 = u1 

 4352/256 = u1

17 = u1

So the first term is 17. The division might seems tricky, but as long as you know basic division, it should be fine.

If you get that the nth term is 272, r is 4 and first term is 17, what term number is the nth term?

The first thing we do is to plug in what we know into the formula:

272 = 17 × 4n-1.

divide both sides by 17

272/17 = 4n-1

16 = 4n-1.

44 = 4n-1.

4 = n-1

n = 3

So 272 is the third term in the sequnce. The trick when finding the term number n in a geometric sequnce, is to get numbers on both sides to to the same base number. Since you can't really do equations with exponents, you have to have the same base number on both side so that you can use the numbers in the exponents in their own equation. That way, you can find the missing value in an exponent.

The last thing is if you're missing the ratio in the sequence. To calculate the ratio of a sequence, you normally take 2 consecutive terms and divide the one with the greatest term number with the other one. For example, using the sequence used eariler:

r = u2/u1.

r = 68/17

r = 4

You can also do

r = u5/u4.

r = 4352/1088

r = 1088.

What you cannot do however, is find the ratio between two terms with more than 1 in difference of term number. So

r = u5/u3.

r = 4352/272

r = 16

does not work.

Instead, if given the 5th is 4352 and the 1st term is 17, you still just plus in the values:

4352 = 17 × r5-1.

divide both sides by 17

256 = r4.

44 = r4

r = 4

Although you might get a different scenario where you are given, the 5th term is 4352 and the 3rd  is 272. In this case you can do one of two things. You can either say that the 3rd term is now the first term, meaning the 5th is the 3rd, or you could leave it as is, but do n-3 instead of n-1. Either way you get the correct answer. 

4352 = 272 × r3-1 

16 = r2 

42  = r2 

r = 4

or

4352 = 272 × r5-3 

16 = r2 

42  = r2

r = 4.

As seen above, both methods are quite similar and both lead to the same answer.

The first thing we have to do is find the ratio. So

u4 = u2 × r4-2.

128.5 = 2056 × r2.

128.5/2056 = r2.

r = 0.25

Now that we have the ratio, we can move on to finding the 7th term. Thusly, we plug the numbers into the general formula:

u7 = 128.5 × 0.257-4

(I use the 4th term here becuase we don't have the 1st and since the ratio is under 0, the 4th term is closer in value to 7th term than the 1st or 2nd would be)

u7 = 128.5 × 0.015625

u7 = 2.00 

(rounded to 2 decimal places)