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Pyramid Numbers Problem Solved

Updated on July 2, 2013

Pyramid Numbers

Section 1: Symbols and Conventions

The following symbols and conventions will be used throughout the entire report.

Section 2: numerator

If we look at the numerators, 3,6,10,15. We can see a pattern, which is, the difference between the numbers, is increasing by 1.

6 - 3 = 3

10 - 6 = 4

15 - 10 = 15

This gives as an airthematic sequence and we can guess the next number of the numerator. Which is going to be (previous term + 6)

15+6 = 21; because all the inner and outer terms are the same in case of the numerator. Hence we can say that the numerator for the 6th row is 21, except at position 0 and the final term, where the nemerator is 1.

Section 2A: Relationship between numerator and row number

Section 3: Finding the next two rows

We will divide the problem into three parts to get the terms for the next two rows.

Section 3A: Numerator

We can reffer back to the section 2, where we already found out a pattern to get the numerator as rows go on. The difference between the numbers increases by 1. Since the numerator is same for the inner and outter terms, we can say that the numerator for row 6 is 21 and for row 7 it is 28.

Section 3B: Denominator (Outer Terms)

If we look at the denominator for outer terms, 1,2,4,7,11. We can see a pattern, which is, the difference between the numbers, is increasing by 1 again.

2 - 1 = 1

4 - 2 = 2

7 - 4 = 3

11 - 7 = 4

This gives as an airthematic sequence and we can guess the next denominators.

Section 3C: Denominator (Inner Terms)

We note that inner denominators (1 < r < n-1 ) decrease symmetrically from outer denominators ( r =1 and r = n-1 ) as we approach the centre of a row.

As we move down the rows, the difference between the denominators increases. Using this pattern we can find the inner denominator. When we know the outer denominator, we can simply subtract the difference from the denominator and move towards the centre.

We can see this in the diagram below:

Section 4: Finding the general formula

In this section, we will find foumulae for:

  1. The numerators of the terms of each row
  2. the denominators of the outer terms of each row
  3. the difference between the denominators of outer and inner terms
  4. the denominators of the inner terms
  5. the general formula for any term in the pyramid

Section 4A: Obtaining an Expression for Nn

Let the sequence made of outer numerators ( r = 1 and r = n-1 ) be denoted by 'Nn', let the difference between every subsequent terms of this sequence be denoted by 'dn' and the difference between subsequent terms of 'dn' be denoted by 'd2n' .

Using these conventions above, we obtain the following table:

Thus, the sequence dn is arithmetic progression with first term, d1 = 2and common difference d = 1 . We denote the arithmetic series associated with dn as Sn .

If we look at the numerators again, we observe:

Hence we can calculate as follows:

Section 4B: Obtaining an expression for the Outer and Inner denominators

Let the sequence made of outer denominators (r = 1 and r = n-1 ) be denoted by Dn, let the difference between every subsequent terms of this sequence be denoted by Dn and the difference between subsequent terms of dn be denoted by d2n .

Using these conventions above, we obtain the following table:

Thus, the sequence is arithmetic progression with first term, d1 = 1 and common difference d = 1. We denote the arithmetic series associated with dn as Sn .

If we look at the denominators Dn again, we observe:

Hence we can calculate as follows:

Section 4C: Obtaining an expression for the difference between Outer and Inner denominators

We note that inner denominators ( 1 < r < n-1 ) decrease symmetrically from outer denominators ( r = 1 and r = n -1 ) as we approach the centre of a row.

As we move down the rows, the difference between the denominators increases. Let Rn be the difference between the denominators in row n. We observer:

R1 = R2 = R3 = 0

R4 = 1 = 4 - 3

R5 = 2 = 5 - 3

Thus we use the following formula for :

Rn = n-3, n (greater or equal to) 4

Section 4D.1: Obtaining an expression for Denominators for row and

In the above range, the terms within the row decrease arithmetically with a common difference of -Rn (see Section 4C.1 for calculation of Rn ). The first term of this progression is given by:

Dn(1) = Dn'

where Dn was calculated in Section 4C. Subsequently, the denominators can be calculated as follows:

Dn(2) = Dn - Rn

Dn(3) = Dn - 2Rn

Dn(4) = Dn - 3Rn

and so on...

Section 4D.2: Obtaining an expression for denominators for row n and (n/2) < r < n-1

In the above range, the terms within the row increase arithmetically from the centre, with a common difference of Rn (see Section 4C.1 for calculation of Rn ).

But since we do not know the middle term of the row, we do not know the first term of the progression. But if we start from the term at r = n - 1 , we note that the terms again decrease arithmetically towards the centre, with a common difference of -Rn . Thus, we are going to focus on this arithmetic progression.

To analyse this progression, we define a new index, k for the terms such that

when , r = n-1, k =1

when , r = n-2, k =2

when , r = n-3, k =3

and so on.

Thus we get a relationship between and as follows:

r = n - k

=> k = n - r

Now, if we use as index, we get exactly the same case as in Section 4C.2 and hence for the range (n/2) < r < n-1, we can write:

Dn(k) = Dn - (k-1)Rn

By substituting k = n - r , we obtain:

Dn(r) = Dn - (n - r -1)Rn


Section 4E: Obtaining the final formula

From the results in section 4B, we have:

Finally any general term in the pyramid can be obtained as follows:

En(r)= { (Nn(r)) / (Dn(r)) }

Section 5: Testing validity of the general formula

To test the validity of the general formula, we will calculate rows 8 and 9 and check it against the values returned by the formula. By following the discussions in sections 2 and 3, we can calculate by hand the terms for row 8 and 9. They are presented in the table below.

Section 6: Scopes and Limitations

The formula presented in section 4E applies for all positive integer values of and for all non-negative integer values of r . Thus the scope of the solution is

The only limitation this formula presents is that it is not valid for negative values of n and r .

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