Questions: Use inductive reasoning to find a pattern for the answers. Then use the pattern to guess the result of the final calculation, and perform the operation to see if your answer is correct. 999,999 × 1 = 0,999,999 999,999 × 2 = 1,999,998 999,999 × 3 = 2,999,997 ... 999,999 × 9 = □

Use inductive reasoning to find a pattern for the answers. Then use the pattern to guess the result of the final calculation, and perform the operation to see if your answer is correct. 999,999 × 1 = 0,999,999 999,999 × 2 = 1,999,998 999,999 × 3 = 2,999,997 ... 999,999 × 9 = □
Transcript text: Use inductive reasoning to find a pattern for the answers. Then use the pattern to guess the result of the final calculation, and perform the operation to see if your answer is correct. \[ \begin{array}{c} 999,999 \times 1=0,999,999 \\ 999,999 \times 2=1,999,998 \\ 999,999 \times 3=2,999,997 \\ \ldots \\ 999,999 \times 9=\square \end{array} \]
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Solution

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Solution Steps

To find the pattern, observe the results of the multiplications:

  • \(999,999 \times 1 = 0,999,999\)
  • \(999,999 \times 2 = 1,999,998\)
  • \(999,999 \times 3 = 2,999,997\)

Notice that the result of each multiplication is \(999,999 \times n = (n-1)999,999 + (10^6 - n)\). Using this pattern, we can predict the result for \(999,999 \times 9\).

Step 1: Identify the Pattern

We observe the results of the multiplications: \[ \begin{array}{c} 999,999 \times 1 = 0,999,999 \\ 999,999 \times 2 = 1,999,998 \\ 999,999 \times 3 = 2,999,997 \\ \end{array} \]

From this, we can deduce a pattern: \[ 999,999 \times n = (n-1) \times 1,000,000 + (1,000,000 - n) \]

Step 2: Apply the Pattern

Using the pattern, we calculate the result for \(999,999 \times 9\): \[ 999,999 \times 9 = (9-1) \times 1,000,000 + (1,000,000 - 9) \]

Step 3: Perform the Calculation

\[ 999,999 \times 9 = 8 \times 1,000,000 + 991,000 = 8,999,991 \]

Final Answer

\(\boxed{8,999,991}\)

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