Sequence question to do with tile patterns

[Natasha1]

Homework Statement

There are patterns made up of square tiles which I can not draw but only describe...

Pattern 1 has 1 grey tile in centre and 8 dotted tiles around it
Pattern 2 has 4 grey tiles in centre and 12 dotted tiles aound it
Pattern 3 has 9 grey tiles in centre and 16 dotted tiles around it
Pattern 4 has 16 grey tiles in centre and 20 dotted tiles around it
and so on

Homework Equations

So the table looks like this

Pattern number 1, 2, 3, 4, ...
Number of grey tiles 1, 4, 9, 16, ...
Number of dotted tiles 8, 12, 16, 20, ...


3. Work out what the total number of tiles there are in pattern 11

So I got for number of grey tiles pattern number 11 squared = 121 and nth term for the number of dotted tiles is 4n+4 so for n=11 we get 48

Hence total number of tiles is 121+48= 169

4. What is the number of the pattern in which the total number of tiles is equal to tsquared?

As the grey tiles are already equal to tsquared for any pattern number t, so any additional tiles e.g. any dotted tiles would just add more to tsquared. So, the only number of the pattern in which the total number of tiles is equal to tsquared must be pattern 0 (zero).

Am I correct? If not, explain why?

Is this correct?

 

[Natasha1]

I should have put this in the maths homework help, rather than the physics. I am not sure how to move it across, sorry...

 

[Curious3141]

Homework Statement

There are patterns made up of square tiles which I can not draw but only describe...

Pattern 1 has 1 grey tile in centre and 8 dotted tiles around it
Pattern 2 has 4 grey tiles in centre and 12 dotted tiles aound it
Pattern 3 has 9 grey tiles in centre and 16 dotted tiles around it
Pattern 4 has 16 grey tiles in centre and 20 dotted tiles around it
and so on

Homework Equations

So the table looks like this

Pattern number 1, 2, 3, 4, ...
Number of grey tiles 1, 4, 9, 16, ...
Number of dotted tiles 8, 12, 16, 20, ...3. Work out what the total number of tiles there are in pattern 11

So I got for number of grey tiles pattern number 11 squared = 121 and nth term for the number of dotted tiles is 4n+4 so for n=11 we get 48

Hence total number of tiles is 121+48= 169

4. What is the number of the pattern in which the total number of tiles is equal to tsquared?

As the grey tiles are already equal to tsquared for any pattern number t, so any additional tiles e.g. any dotted tiles would just add more to tsquared. So, the only number of the pattern in which the total number of tiles is equal to tsquared must be pattern 0 (zero).

Am I correct? If not, explain why?

Is this correct?

The first part seems OK. The mathematical relationship for the total number of tiles can be expressed as $T_n = n^2+4n+4$, where n is the "pattern number" as you referred to it - more formally, this number is referred to as an index.

The second part seems to ask for which values of n is the total no. of tiles a perfect square (I don't think they meant the total number is the square of that particular index or "pattern number"). Well, does the expression for $T_n$ remind you of anything?

EDIT: you can't move the thread, but you can PM a mentor to help you to move it. I suggest you do this rather than try to create a duplicate thread in a separate forum.

 

[Natasha1]

The quadratic has only two solutions which are -2. As -2 is not part of the index or 'pattern number', I was correct in saying the only possible 'pattern number' is 0 (zero). Am I correct please?

 

[haruspex]

4. What is the number of the pattern in which the total number of tiles is equal to tsquared?

As the grey tiles are already equal to tsquared for any pattern number t, so any additional tiles e.g. any dotted tiles would just add more to tsquared. So, the only number of the pattern in which the total number of tiles is equal to tsquared must be pattern 0 (zero).

You are assuming that the answer is t. Suppose the answer is n. How many tiles in total in pattern n? What do you get if you set that equal to t²?

 

[Natasha1]

I really don't get this sadly

 

[Curious3141]

I really don't get this sadly

Do you get what I said about $T_n = n^2 + 4n + 4$, where $T_n$ represents the total number of tiles for index (what you call "pattern number" n)?

If so, try listing the values of $T_n$ for n = 1,2,3,... maybe until 10 or so. Do you see that all the numbers share a common property?

Make a table with 3 columns. The first column is n, second is $T_n$, and the third is $n+2$. Can you now deduce what the pattern is? How do you prove it?

 

[Natasha1]

n = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
Tn = 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 (they are square numbers)
n+2 = 3, 4, 5, 6, 7, 8, 9, 10, 11, 12

The pattern is (n+2)squared. But how can this help me answer question 4. What is the number of the pattern in which the total number of tiles is equal to tsquared? Is the answer all of them?

I am confused with the wording of the question.

Is the number of the pattern n+2? Is that actually a number?

 

[Curious3141]

n = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
Tn = 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 (they are square numbers)
n+2 = 3, 4, 5, 6, 7, 8, 9, 10, 11, 12

The pattern is (n+2)squared. But how can this help me answer question 4. What is the number of the pattern in which the total number of tiles is equal to tsquared? Is the answer all of them?

I am confused with the wording of the question.

If that's exactly how the question is worded, then I'm confused too. If that's not how it's typed out *exactly*, you should probably reproduce the actual question to clarify.

What I can tell you is that every single total number of tiles in the sequence is a perfect square, and it's equal to to $(n+2)^2$, where n is the index of the sequence.

Is the number of the pattern n+2? Is that actually a number?

I'm confused by what you mean by this. n is the index of the tile sequence. An index is a number signifying the place in an ordered pattern. So n = 1 means "1st number in sequence", and so forth. You can represent the actual values in the sequence (the total number of tiles, in this instance) by a formula that depends on n. In this case, that formula is $T_n = (n+2)^2$.

This is just one sequence. There are an infinite possible number of sequences, e.g. sequence of natural numbers: 1,2,3,4... (where the values are equal to the indices, i.e. $T_n = n$), an arithmetic sequence like 2, 5, 8, 11, ... (where the value is given by $T_n = 3n-1$), a geometric sequence like 2, 4, 8, 16, ... (where the value is given by $T_n = 2^n$), and many, many other sorts, some of which are downright esoteric. Sequences can either be finite (limited to a maximum value of the index n, which needs to be specified), or infinite (go on forever, so that n can be infinitely large).

Is any of what I wrote confusing?

 

[Natasha1]

A little but I get it. The question is exactly written the way I typed it e.g. 4. What is the number of the pattern in which the total number of tiles is equal to tsquared?

 

[Curious3141]

A little but I get it. The question is exactly written the way I typed it e.g. 4. What is the number of the pattern in which the total number of tiles is equal to tsquared?

Did the question setter previously define t (in $t^2$)? If not, then the answer is "all of them" - every total number of tiles is a perfect square that can be represented as $t^2$, where t is an integer.

I think your confusion stems from you having assumed that t represents the "pattern number" (index of the sequence). If they didn't actually state that, there's no reason you should think so.

 

[Natasha1]

The answer from is t-2 apparently! Can someone explain why?

 

[Curious3141]

The answer from is t-2 apparently! Can someone explain why?

I suggest that you reproduce the question exactly, e.g. by scanning it into an image attachment, before proceeding any further. Too much ambiguity in what t means, etc.

 

[haruspex]

What is the number of the pattern in which the total number of tiles is equal to tsquared?

The answer will be in terms of t.
You already deduced that if the index is n then the number of tiles is (n+2)². All the question is asking you to do is to run this relationship backwards: if the number of tiles is t², what is the index? So suppose the answer is n. You know how many tiles a pattern of index n will have, so what equation can you write relating n and t?

 

[Curious3141]

The answer will be in terms of t.
You already deduced that if the index is n then the number of tiles is (n+2)². All the question is asking you to do is to run this relationship backwards: if the number of tiles is t², what is the index? So suppose the answer is n. You know how many tiles a pattern of index n will have, so what equation can you write relating n and t?

Honestly, this should be a very easy question (if this is what Natasha meant), but I was thrown by this ambiguity:

4. What is the number of the pattern in which the total number of tiles is equal to tsquared?

As the grey tiles are already equal to tsquared for any pattern number t,

The "pattern number" is referred to, variously, as "n" and then "t". If that's the same "t" as in "t²", then the answer is "never" because that involves the equation $(t+2)^2 = t^2$ giving an inadmissible solution.

So, the only question is whether "t" defined as the "pattern number" (index) by the question or whether Natasha assumed that t represented the pattern number. Your interpretation is reasonable, but I think it's better the thread starter clarifies the question, if only so that she properly understands it (because right now, I doubt she fully understands what's going on).