Question

The Number Pyramid ProblemA number pattern is shown above. A.) what is the difference between the last number is row 6 and the first number in row 6? Show or explain your work. B.) write and equation that can be used to find D the difference between the last number and the first number, for any row “R”. Show or explain your work. C.) What is the first number in row 11? What is the last number is row 11? Show or explain your work. D.) What is the number in row 1? Show or explain your work.

Answer 1

Let us first find the pattern in the difference between the first and the last number for a given row.

Row 2: 10 - 8 = 2

Row 3: 19 - 15 = 4

Row 4: 30 - 24 = 6

Row 5: 43 - 35 = 8

Do you see a pattern here?

There is a constant difference of 2 between each row

Part A:

What is the difference between the last number is row 6 and the first number in row 6?

Using the above pattern, the difference must be 10

Part B:

Write an equation that can be used to find D the difference between the last number and the first number, for any row “R”

`D_R=(R-1)\cdot2`

Where R is the row number.

Let us verify if the above equation works

`\begin{gathered} D_4=(4-1)\cdot2=3\cdot2=6 \\ D_5=(5-1)\cdot2=4\cdot2=8 \end{gathered}`

As you can see, the equation works fine.

Part C:

What is the first number in row 11? What is the last number is row 11? Show or explain your work.

As you can see from the pattern, the difference between the last number and the first number of the next row is 5

Row 6:

First number = 43 + 5 = 47

Last number = 47 + 10 = 57

Row 7:

First number = 57 + 5 = 62

Last number = 62 + 12 = 74

Row 8:

First number = 74 + 5 = 79

Last number = 79 + 14 = 93

Row 9:

First number = 93 + 5 = 98

Last number = 98 + 16 = 114

Row 10:

First number = 114 + 5 = 119

Last number = 119 + 18 = 137

Row 11:

First number = 137 + 5 = 142

Last number = 142 + 20 = 162

Therefore, the first number in row 11 is 142 and the last number is 162

Part D:

What is the number in row 1?

The difference between the number in row 1 and the first number in row 2 must be 5

So, 8 - 5 = 3

The number in row 1 must be 3