Question

I need help with this please! I want someone to go through the steps with me, not tell me the answer. Thanks~! ^_^

Answer 1

@zepdrix can you help?

Answer 2

What's up? :O
Confused on this honey business?

Answer 3

XD Yes. My first time with sequences. lolol

Answer 4

\[\large\rm f(\color{indianred}{n})=6\color{indianred}{n}\]Recognize that the explicit function is pretty much exactly like all other function stuff you've dealt with up to this point. You plug in a value for n and try to determine the output of the function.

So for n=7, ring 7,\[\large\rm f(\color{indianred}{7})=6(\color{indianred}{7})\]how many do you get?

Answer 5

42?

Answer 6

Ok good, so our explicit formula was pretty straight forward.
42 cells in the 7th ring.

Answer 7

For our recursive function it's a little bit different.
\(\large\rm \color{royalblue}{f(1)=6}\)
This thing in blue is telling us that when n=1, the function = 6.
In other words, ring 1 has 6 cells.

We'll need this information for all other calculations that we do.

Answer 8

So here is our recursive relationship,\[\large\rm f(\color{indianred}{n})=\color{royalblue}{f(1)}+f(\color{indianred}{n}-1)\]

Answer 9

\[\large\rm f(\color{indianred}{n})=\color{royalblue}{6}+f(\color{indianred}{n}-1)\]

Answer 10

There is no sense in plugging in n=1,
we already know what's happening in ring 1.

So let's try to find out how many cells are in ring 2.
\[\large\rm f(\color{indianred}{2})=\color{royalblue}{6}+f(\color{indianred}{2}-1)\]This is how recursive works,
we can't just jump right to n=6,
we have to calculate every n leading up to n=6 first.

Answer 11

Doing the subtraction, 2-1 = 1,\[\large\rm f(\color{indianred}{2})=\color{royalblue}{6}+f(1)\]Hmm that's interesting, we know something about f(1) don't we?

Answer 12

I'm not to sure. What is it? I got everything else.

Answer 13

We had this at the very start of the problem,\[\large\rm \color{royalblue}{f(1)=6}\]

Answer 14

Alright.

Answer 15

So we're plugging in n=2,
this tells us that we're looking at ring 2,

and everything on the right side of the equals is telling us how many cells are in ring 2.