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Mathematics Borders

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Introduction

Level 8

I will now express the generalization formula in terms of sigma notation using square numbers.

Below, I will show the sequence of cubes need to make a 3D cross shape through the help of level 7 in the form of a cross shape. This will be presented in a tabular manner

 Shape Number of cubes Sigma Notation 1 1 12+ 0 2 7 12+ 22 + 2 3 25 12 + 22 + 32 + 11 4 63 12+ 22 + 32 + 42 + 33 5 129 12+ 22 + 32 + 42 +52 + 74 6 231 12+ 22 + 32 + 42 + 52 + 62 +140

6Σ =r2

r = 1

Since I have found the formula

= Un = 4 n3 - 2n2 + 8 - 1

1. 3

This

Middle

2nd difference →    5                     7                     9                  11

3rd difference →            2                   2                      2

The 3rd difference is denoted by the term 6a.

Therefore in order to find a, I will replace the 3rd difference value

6a = 2

a = 2 = 1

6    3

By the given formula, Un = an3 + bn2 +cn + d, I will now be able to arrive to the generalized formula.

• If n=1

U1 = an3 + bn2 +cn + d

1 = 1 (1)3 + b(1)2 + c(1) + d

3

= b + c + d = 2       [equation 1]

3

• If n = 2

U2 = 1 (2)3 + b(2)2 + c(2) + d

3

5 = 8+ 4b + 2c + d

3

= 4b + 2c + d = 7    [equation 2]

3

• If n=3

U3 = 1 (3)3 + b(3)2 + c(3) + d

3

14 = 9 + 9b + 3c + d

= 9b + 3c + d = 5      [equation 3]

[equation 2 – equation 1]

4b + 2c + d = 7

3

b + c + d = 2

3

= 3b + c = 5/3     [equation3]

• [equation 3 – equation 2]

9b + 3c + d = 5

4b + 2c + d = 7

3

= 5b + c = 8        [equation 5]

3

• [equation 5 – equation 4]

5b + c = 8

3

3b + c = 5

3

= 2b = 1

= b = ½

Conclusion

• [equation 3 – equation 2]

9b + 3c + d = -16

4b + 2c + d = -6

= 5b + c = -10                [equation 5]

• [equation 5 – equation 4]

5b + c = -10

3b + c = -5

= 2b = -5

= b = -5

2

In order to find c I will replace the value b = -5 in equation 5

2

5 * -5+ c = -10

2

-25 + c = -10

2

c = -10 + 25

2

c = -20 + 25

2

c = 5

2

In order to find d I will now replace b and c in equation 1

-5+ 5 + d = - 1

2    2

0 + d = - 1

Therefore

a=1

b= -5

2

c=5

2

d= -1

Therefore generalized formula is = n3 – 5n2 + 5n – 1      [formula 2]

2         2

After obtaining both the formulas, I will now be adding formula 1 and formula 2 to justify the formula I got in level 7.

1n3 + 1n2 + 1n

3        2        6

1n3 – 5n2 + 5n – 1

2        2

= 4n3– 2n2 + 8n – 1

3                3

Thus I do obtain the formula which I had obtained in level 7. This thus justifies that my formula is right.

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