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Analyze patterns with two rules

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5170595
Continue both number patterns logically, and find every sum. 1. \(210{,}000 + 320{,}000 = \underline{\hspace{1cm}}\) 2. \(220{,}000 + 330{,}000 = \underline{\hspace{1cm}}\) 3. \(230{,}000 + 340{,}000 = \underline{\hspace{1cm}}\) 4. \(\underline{\hspace{1cm}} + \underline{\hspace{1cm}} = \underline{\hspace{1cm}}\) 5. \(\underline{\hspace{1cm}} + \underline{\hspace{1cm}} = \underline{\hspace{1cm}}\)

Hints

- Determine the change in each addend separately. - Apply both rules to create the next pair of addends. - When both addends increase by the same amount, consider how much the sum increases.

Solution

1. The first addend increases by \(10{,}000\) each time, and the second addend also increases by \(10{,}000\) each time. 2. The first three sums are \(530{,}000\), \(550{,}000\), and \(570{,}000\). 3. Continue the two rules: \(240{,}000 + 350{,}000 = 590{,}000\), then \(250{,}000 + 360{,}000 = 610{,}000\).

Answer

1. \(530{,}000\) 2. \(550{,}000\) 3. \(570{,}000\) 4. \(240{,}000 + 350{,}000 = 590{,}000\) 5. \(250{,}000 + 360{,}000 = 610{,}000\)
5203905
Lucas builds staircases from cubes. Each step is two cubes wide. He records his observations in a table: <table> <tr> <td>Staircase height</td> <td>1</td> <td>2</td> <td>3</td> </tr> <tr> <td>Cubes in the bottom row</td> <td>2</td> <td>4</td> <td>6</td> </tr> <tr> <td>Total number of cubes</td> <td>2</td> <td>6</td> <td>12</td> </tr> </table> a) For a staircase with height \(10\), how many cubes are in the bottom row, and how many cubes are needed altogether? b) Describe how to find the bottom-row count and total cube count directly from a staircase height \(h\).

Hints

- Compare the height with the bottom-row count. - Add the even row counts from top to bottom. - Look for a direct rule that uses the height and the next whole number.

Solution

1. The bottom row has twice as many cubes as the height. For height \(10\), \(10\times2=20\) cubes are in the bottom row. 2. The total is \(2+4+6+8+10+12+14+16+18+20=110\). 3. For height \(h\), the bottom-row count is \(2\times h\). The total cube count is \(h\times(h+1)\). For \(h=10\), \(10\times11=110\).

Answer

a) Bottom row: \(20\) cubes; total: \(110\) cubes b) Bottom row: \(2\times h\); total: \(h\times(h+1)\)

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