Study the following pattern:
[tex]$\begin{array}{l}
84 \times 99=8,316 \\
84 \times 999=83,916 \\
84 \times 9,999=8,39,916
\end{array}$[/tex]
Write the next seven steps in the pattern.
Answer (1)
The pattern involves multiplying 84 by numbers consisting of only 9s.
The result follows the pattern 8, 3, a sequence of 9s, and 16.
The number of 9s in the result is two less than the number of 9s in the multiplier.
The next seven steps are: 84 × 99999 = 8399916 , 84 × 999999 = 83999916 , 84 × 9999999 = 839999916 , 84 × 99999999 = 8399999916 , 84 × 999999999 = 83999999916 , 84 × 9999999999 = 839999999916 , 84 × 99999999999 = 8399999999916 .
8399916 , 83999916 , 839999916 , 8399999916 , 83999999916 , 839999999916 , 8399999999916
Explanation
Analyzing the Pattern We are given a pattern of multiplication involving the number 84 and numbers consisting of only 9s. We need to extend this pattern for the next seven steps. The pattern observed is that when 84 is multiplied by a number with n 9s, the result is a number that starts with 8, followed by n − 2 9s, and ends with 16. The digit 3 appears after the 8.
Extending the Pattern Based on the pattern, we can write the next seven steps as follows:
84 × 99999 = 8399916
84 × 999999 = 83999916
84 × 9999999 = 839999916
84 × 99999999 = 8399999916
84 × 999999999 = 83999999916
84 × 9999999999 = 839999999916
84 × 99999999999 = 8399999999916
Final Results Therefore, the next seven steps in the pattern are:
84 × 99999 = 8399916 84 × 999999 = 83999916 84 × 9999999 = 839999916 84 × 99999999 = 8399999916 84 × 999999999 = 83999999916 84 × 9999999999 = 839999999916 84 × 99999999999 = 8399999999916
Examples
This pattern can be useful in quick mental calculations or in understanding number patterns. For example, if you need to quickly estimate the product of 84 and a number like 99999, you can use this pattern to arrive at the answer without performing the full multiplication. Understanding such patterns helps in developing a better intuition for numbers and their relationships, which is valuable in various fields like finance, engineering, and computer science.
