Question

A group of students is arranging squares into layers to create a project. The first layer has 4 squares. The second layer has 8 squares. Which formula represents an arithmetic explicit formula to determine the number of squares in each layer?

f(1) = 4; f(n) = 4 ⋅ d(n − 1), n > 0
f(1) = 4; f(n) = 4 ⋅ d(n + 1), n > 0
f(1) = 4; f(n) = 4 + d(n + 1), n > 0
f(1) = 4; f(n) = 4 + d(n − 1), n > 0

Answer 1

@darkknight

Answer 2

okay so these are our answer choices
f(1) = 4; f(n) = 4 ⋅ d(n − 1), n > 0
f(1) = 4; f(n) = 4 ⋅ d(n + 1), n > 0
f(1) = 4; f(n) = 4 + d(n + 1), n > 0
f(1) = 4; f(n) = 4 + d(n − 1), n > 0

Now Lets see how these work out,
First equation, if we plug in 1 for 4, what do we get? Well lets see, it would evaluate to 4 * d(1-1) = 4* d(0) = 0. But this makes no sense, because f(1) = 4 not 0, therefore the first answer choice is incorrect,

Answer 3

actually hold up, lemme double check that real quick

Answer 4

Ight.

Answer 5

Actually thinking about it, the easiest way to solve this problem is to plug in points, and see if this satisfies the correct arithmetic formula, so lets do that

Answer 6

So basically do you know the arithmetic explicit formula? how it is written in its standard form?

Answer 7

i think it is f(n) = f(1) + d(n-1).....

Answer 8

Oh actually before I get to what I was saying, this is an arithmetic explicit formula, not geometric, so it can't be in the form of something times d(...) it has to be something plus d(...) So we can eliminate the first 2 answer choices like that

Answer 9

ah ok

Answer 10

Okay Lets see how we can write this, 4 + d(n-1)
When n =1 then
4 + d(1-1) = 4+d(0) = 4,
And this is correct, because f(1) is 4.
Lets assign d a value of 4, because we are adding 4 to every layer.
Now we also know that f(2) = 8
Lets see if that works
4+ d(2-1) = 8
4+ d(1) = 8
d(1) = 4
4(1) = 4 THIS is CORRECT
Lets try plugging in 3 for n, we know that f(3) = 12
So than
4+ d(3-1) = 12
4+d(2) = 12
d(2) = 8
4(2) = 8 THIS IS CORRECT AS WELL
Since this equation satisfies all the values we plugged in for n, this is correct.

Answer 11

lmk, if anything doesn't make sense : )

Answer 12

Everything doesn't make sense.....

Answer 13

Like then which one is correct?

Answer 14

oof, hold on. So basically lemme copy-paste what I wrote above\(\color{#0cbb34}{\text{Originally Posted by}}\) @darkknight
Okay Lets see how we can write this, 4 + d(n-1)
When n =1 then
4 + d(1-1) = 4+d(0) = 4,
And this is correct, because f(1) is 4.
Lets assign d a value of 4, because we are adding 4 to every layer.
Now we also know that f(2) = 8
Lets see if that works
4+ d(2-1) = 8
4+ d(1) = 8
d(1) = 4
4(1) = 4 THIS is CORRECT
Lets try plugging in 3 for n, we know that f(3) = 12
So than
4+ d(3-1) = 12
4+d(2) = 12
d(2) = 8
4(2) = 8 THIS IS CORRECT AS WELL
Since this equation satisfies all the values we plugged in for n, this is correct.
\(\color{#0cbb34}{\text{End of Quote}}\)
Basically that is what I said, I took the assumption that the last function was the correct one and then I plugged in values for n to see if they satisfy. Like we had 1,4) and (2,8) So I plugged 1 and 2 for n and saw if they returned 4 and 8 back respectively

Answer 15

Basically what I did is that I assigned d = 4. I did this because for every row, the squares increased by 4,

Answer 16

So now
4 + 4(n-1) = ?
If we plug a value into this we should get the correct output, for example if I plug 1 for n, (remember that on the first row there are 4 squares) we get 4
4+4(1-1) = 4 + 4(0) = 4
and likewise if we plug in 2 for n
4+4(2-1) = 4+ 4(1) = 8 (correct because on row 2 we had 8 squares)

Answer 17

Yes it is.

Answer 18

to sum everything up, we saw that everytime another row is added, we see 4 more squares, that is why I said d = 4. Now we are finding an arithmetic explicit sequence so the only valid functions of the 4 answer choices are the last 2, because the first 2 are geometric explicit sequences. So that is why we eliminated the last 2 choices.
Now I substituted values into the last equation to see if it satisfies the values
"So now
4 + 4(n-1) = ?
If we plug a value into this we should get the correct output, for example if I plug 1 for n, (remember that on the first row there are 4 squares) we get 4
4+4(1-1) = 4 + 4(0) = 4
and likewise if we plug in 2 for n
4+4(2-1) = 4+ 4(1) = 8 (correct because on row 2 we had 8 squares)"
If the left side equals the right side, then the function is correct, in the case of option choice 3
4+ 4(n+1) = ?
4+4(1+1) = 12
In our first row we had 4 squares, not 12. That is why the 3rd answer choice is incorrect
This should help a lot, @supie

Answer 19

Yeah, that helps a lot, thanks @DARKKNIGHT

Answer 20

np