Write a formula for this sequence: 1/5, 3/5, 1, 1 2/5..

Question

Write a formula for this sequence:  1/5, 3/5, 1, 1 2/5..

kainui · Answer

Hmm, is there some way we can try to find out the difference between what happens at each step to see if there's some pattern there?

anonymous · Answer

thats all i have. /:

kainui · Answer

How much did they add between the first number and the second number?

anonymous · Answer

2

kainui · Answer

Good! Although don't forget those 5's at the bottom, it doesn't really change much except they added 2/5.

kainui · Answer

So how much did they add to get from the second number to the third number?

anonymous · Answer

2/5?

kainui · Answer

Yeah! After all, 3 fifths added to 2 fifths is 5 fifths, which makes sense that would be 1.

Alright, now let's see if we can make a formula. Try some stuff out and don't be afraid to fail!

anonymous · Answer

but 2/5's added to 1 doesnt match with 2/5's

kainui · Answer

The last number is the mixed number $$\Huge 1 \frac{2}{5}$$ correct?

anonymous · Answer

i did not see the 1 infront, but yes it is

kainui · Answer

Ahh, well now that you see it, it makes sense right?

anonymous · Answer

no.

kainui · Answer

What term is not 2/5 greater than the previous term?

anonymous · Answer

wait yes it makes sense

kainui · Answer

Ok cool. So now every term is really just the first term with a bunch of extra 2/5 added to it right?

We could rewrite the sequence as: $$\Large \frac{1}{5},\ \ \frac{1}{5}+\frac{2}{5}, \ \ \frac{1}{5}+\frac{2}{5}+\frac{2}{5}, \ \frac{1}{5}+\frac{2}{5}+\frac{2}{5}+\frac{2}{5}, \ \ etc$$

See what I mean? But is there a better way to do this?

anonymous · Answer

im not sure

kainui · Answer

If I give you 3+3+3+3 would you say that's the same as 4*3 since there are 4 threes?

anonymous · Answer

yes

kainui · Answer

Let's look at a slightly different sequence then:

1, 4, 7, 10

Notice how we can rewrite this as

1, 1+3, 1+3+3, 1+3+3+3

since each term is just 3 greater than the last But now notice we can combine terms to get:

1+0*3, 1+1*3, 1+2*3, 1+3*3

So if we call each term by a number then we can say that the first term is 0, the next term is 1, the next is 2, etc... then we can write down that

1+3n

will be the formula for this sequence. So if I want the term where n=3 then I plug it in and get

1+3*3=10

and if we want the term before n=3 we plug in n=2

1+2*3=7

and sure enough we are getting the same sequence!

anonymous · Answer

im quite confused

kainui · Answer

By what?