If $1=5, 2=25$, what is 10?
Answer (2)
Observe the pattern: 1 = 5 1 and 2 = 5 2 .
Hypothesize the pattern: n = 5 n .
Apply the pattern to find the value corresponding to 10: 10 = 5 10 .
Calculate 5 10 : 9765625 .
Explanation
Understanding the Problem We are given that 1 = 5 and 2 = 25 . We need to find the value corresponding to 10, following the pattern established by the given equations.
Identifying the Pattern Let's observe the pattern between the numbers on the left-hand side and the right-hand side of the given equations. We can rewrite the given equations as 1 = 5 = 5 1 and 2 = 25 = 5 2 .
Hypothesizing the Pattern We can hypothesize that the pattern is n = 5 n , where n is the number on the left-hand side.
Applying the Pattern Now, let's apply this pattern to find the value corresponding to 10. According to the pattern, 10 = 5 10 .
Calculating the Value Let's calculate 5 10 . The result of the calculation is 5 10 = 9765625 .
Final Answer Therefore, if 1 = 5 and 2 = 25 , then 10 = 9765625 .
Examples
This pattern-based problem can be related to understanding exponential growth in various real-life scenarios. For instance, consider a bacterial colony where the population quintuples with each generation. If the first generation has 5 bacteria and the second has 25, predicting the population at the tenth generation involves calculating 5 10 , which gives you the size of the colony after ten generations. This kind of exponential growth model is also applicable in finance, such as calculating compound interest or investment growth over time.
The value corresponding to 10, given the patterns that 1 = 5 and 2 = 25 , is calculated as 10 = 5 10 . This results in 10 = 9765625 . Thus, if 1 = 5 and 2 = 25 , then 10 = 9765625 .
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