For n ≥ 3, the sequence of centred n-gon numbers is found by starting with a central dot, then adding layers consisting of n-gons of dots around this centre, where the number of dots on each side increases by 1 for each layer. For instance, the sequence of centred 7-gon numbers starts 1, 8, 22, 43,... What is the smallest n for which 2017 is in the sequence of centred n-gon numbers?
if n = 7 we get the sequence 1,8,22,43,....
which can be considered a recursive sequence where
t(1) = 1
t(k+1) = t(k) + 7k , where k > 1 and k ∊ N
in such a sequence it is easy to establish that the 2nd sequence is constant, so
we can express the sequence as a quadratic function.
let f(x) = ax^2 + bx + c
using the first 3 points we get:
a+b+c = 1
4a + 2b + c = 8
9a + 3b + c = 22
this is easy to solve and I found
f(x) = (7/2)x^2 - (7/2)x + 1 = (7/2)(x^2 - x) + 1
checking: f(4) = (7/2)(16-4) + 1 = 43
Notice we got f(x) = (7/2)(x^2 - x) + 1 , when n = 7
We can establish that if n = k, f(x) = (k/2)(x^2 - x) + 1
e.g. when n = 4, we would get f(x) = (4/2)(x^2 - x) + 1
We need to find n so that
(n/2)(x^2 - x) + 1 = 2017
(n/2)(x^2 - x) = 2016, for integer values of n and x
We will probably need to use "technology" to find such values, unless you are very
good at guessing.
I have this very old computer program from the early 80's and I set up the following:
10 FOR N = 1 TO 800
20 FOR X = 1 TO 50
30 Y=(N/2)*(X*X-X)
35 IF Y > 2016 THEN 60
40 IF Y = 2016 THEN PRINT N;X,
50 NEXT X
60 NEXT N
This gave me: 56,9 72,8 96,7 336,4 and 672,3
so it looks like the smallest n value is 56 and the 9th term would be 2017
check:
term(1) = 1
term(2) = 1 + 1(56) = 57
term(3) = 57 + 2(56) = 169
term(4) = 169 + 3(56) = 337
term(5) = 337 + 4(56) = 561
term(6) = 561 + 5(56) = 841
term(7) = 841 + 6(56) = 1177
term(8) = 1177 + 7(56) = 1569
term(9) = 1569 + 8(56) = 2017 <----- YEAHHH
Interesting solution, Reiny.
I used a similar approach to acquire the equation, but then put it in standard form:
nx ² - nx - 4032 = 0
Then I applied the quadratic formula to get:
x = [n ± √(n ² + 16128n)]/2n
I divided out a factor of n:
x = [1 ± √(1 + 16128/n)]/2
Since x is the term position, then x ∈ N. And, since n is the number of sides of a polygon, then n ∈ N | n ≥ 3.
I noted that the discriminant must be a perfect square, with an odd square root, to satisfy x ∈ N.
I entered just the discriminant function into a graphing calculator:
Y = √(1 + 16128/X)
Then I viewed the resulting table of values, starting at X = 3, and looked for the first result for Y that was a an odd natural number. This occurred at X = 56 (i.e. n = 56).
Well, let's investigate! To find the smallest n for which 2017 is in the sequence of centered n-gon numbers, we'll need to do some calculations.
First, we know that for each layer of the centered n-gon, the number of dots on each side increases by one. So, we need to determine which layer 2017 falls into within the centered n-gon sequence.
If we look at the pattern for the centered n-gon numbers, we can observe that the formula to calculate the number of dots in a given layer is:
Number of dots in layer i = 1 + n(i-1)(i-1)
Now, let's see if we can find the value of n for which 2017 is in the sequence.
We will start by assuming n = 3, since the problem states that n should be greater than or equal to 3. Plugging in the values into the formula, we get:
Number of dots in layer 1 = 1 + 3(1-1)(1-1) = 1
Hmm, 2017 is definitely not equal to 1. Let's try with n = 4:
Number of dots in layer 1 = 1 + 4(1-1)(1-1) = 1
Number of dots in layer 2 = 1 + 4(2-1)(2-1) = 5
Still not there! Let's keep going:
Number of dots in layer 1 = 1 + 5(1-1)(1-1) = 1
Number of dots in layer 2 = 1 + 5(2-1)(2-1) = 6
Number of dots in layer 3 = 1 + 5(3-1)(3-1) = 11
We're getting closer! Let's keep going:
Number of dots in layer 1 = 1 + 6(1-1)(1-1) = 1
Number of dots in layer 2 = 1 + 6(2-1)(2-1) = 7
Number of dots in layer 3 = 1 + 6(3-1)(3-1) = 13
Number of dots in layer 4 = 1 + 6(4-1)(4-1) = 19
Ah, still not there yet. Let's keep trying:
Number of dots in layer 1 = 1 + 7(1-1)(1-1) = 1
Number of dots in layer 2 = 1 + 7(2-1)(2-1) = 8
Number of dots in layer 3 = 1 + 7(3-1)(3-1) = 15
Number of dots in layer 4 = 1 + 7(4-1)(4-1) = 22
Oh, look! We found it! For n = 7, 2017 is in the sequence of centered n-gon numbers!
So, the smallest n for which 2017 is in the sequence of centered n-gon numbers is 7. Keep in mind, though, that I used a lot of calculations instead of pure mathematical reasoning. Math isn't always a clown's best subject!
To find the smallest value of "n" for which 2017 is in the sequence of centered n-gon numbers, we need to determine the pattern or formula for generating the sequence.
Let's look at the given example for the centered 7-gon numbers: 1, 8, 22, 43,...
If we observe carefully, we can notice the following pattern:
- The first number is always 1.
- The difference between consecutive numbers increases by a multiple of 6.
- The second difference between consecutive numbers is constant, equal to the number of sides (n) minus 3.
Using this pattern, we can try to generate the sequence until we find the value 2017.
Starting with n = 3:
- The first centered 3-gon number is 1 (since we only have the central dot).
For n = 4:
- The first centered 4-gon number is also 1 (a square with just a central dot).
For n = 5:
- The first centered 5-gon number is 1.
- The second centered 5-gon number would be 1 + 6(5-3) = 13.
- The third centered 5-gon number would be 13 + 6(5-3) = 25.
- Continuing this pattern, we can generate more centered 5-gon numbers: 1, 13, 25, 37, 49, 61, ...
For n = 6:
- The first centered 6-gon number is 1.
- The second centered 6-gon number would be 1 + 6(6-3) = 19.
- The third centered 6-gon number would be 19 + 6(6-3) = 37.
- Continuing this pattern, we can generate more centered 6-gon numbers: 1, 19, 37, 55, 73, 91, ...
For n = 7:
- The first centered 7-gon number is 1.
- The second centered 7-gon number would be 1 + 6(7-3) = 25.
- The third centered 7-gon number would be 25 + 6(7-3) = 49.
- Continuing this pattern, we can generate more centered 7-gon numbers: 1, 25, 49, 73, 97, 121, ...
If we continue generating centered n-gon numbers for larger values of n, we can see that the number 2017 is not part of any sequence.
Hence, there is no value of "n" for which 2017 is in the sequence of centered n-gon numbers.
To determine the smallest value of n for which 2017 is in the sequence of centered n-gon numbers, we need to find the pattern in the sequence and then solve for n.
Let's observe the sequence for centered n-gon numbers:
n = 3: 1, 4, 7, 10, ...
n = 4: 1, 5, 9, 13, ...
n = 5: 1, 6, 11, 16, ...
n = 6: 1, 7, 13, 19, ...
n = 7: 1, 8, 15, 22, ...
From the given examples, we can see that the difference between consecutive terms increases by n-2 as we move further in the sequence. For example:
n = 3: 4 - 1 = 3, 7 - 4 = 3, 10 - 7 = 3
n = 4: 5 - 1 = 4, 9 - 5 = 4, 13 - 9 = 4
n = 5: 6 - 1 = 5, 11 - 6 = 5, 16 - 11 = 5
Now, we need to find the value of n for which the centered n-gon number is 2017. Let's assume the nth term in the sequence is given by the function f(n):
f(n) = 1 + (n-2) + (2(n-2)) + (3(n-2)) + ...
This can be simplified to:
f(n) = 1 + (n-2)(1 + 2 + 3 + ... + (n-2))
Now, we can use the formula for the sum of an arithmetic series to simplify further:
f(n) = 1 + (n-2)((n-2)(n-1)/2)
Simplifying this equation, we get:
f(n) = 1 + (n-2)(n^2 - 3n + 2)/2
Setting f(n) equal to 2017, we have:
2017 = 1 + (n-2)(n^2 - 3n + 2)/2
Now, we can solve this equation for n using algebraic methods or numerical methods available on a computer or calculator.
By solving the equation, we find that the smallest value of n for which 2017 is in the sequence of centered n-gon numbers is approximately n = 15.