To analyze the sequence of coins in the piggy banks, we have the following counts:
- First piggy bank: 1 coin
- Second piggy bank: 4 coins
- Third piggy bank: 12 coins
- Fourth piggy bank: 24 coins
Let's denote the number of coins in each piggy bank as \( a_1, a_2, a_3, a_4 \):
- \( a_1 = 1 \)
- \( a_2 = 4 \)
- \( a_3 = 12 \)
- \( a_4 = 24 \)
Now, let's observe the sequence: \( 1, 4, 12, 24 \).
To see if there's a pattern, we'll look at the ratios between consecutive terms and the differences.
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First, calculate the differences between consecutive terms:
- \( a_2 - a_1 = 4 - 1 = 3 \)
- \( a_3 - a_2 = 12 - 4 = 8 \)
- \( a_4 - a_3 = 24 - 12 = 12 \)
The differences are \( 3, 8, 12 \).
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Next, calculate the differences of these differences:
- \( 8 - 3 = 5 \)
- \( 12 - 8 = 4 \)
The second differences are \( 5, 4 \).
Since the second differences aren't constant, we can also check for a pattern in the ratios:
- \( \frac{a_2}{a_1} = \frac{4}{1} = 4 \)
- \( \frac{a_3}{a_2} = \frac{12}{4} = 3 \)
- \( \frac{a_4}{a_3} = \frac{24}{12} = 2 \)
The ratios also don't suggest a simple geometric sequence.
Another way to investigate this sequence is to look for recurrence relations or a formula.
The numbers \( 1, 4, 12, 24 \) can also be observed as a pattern:
- \( a_n = n! + (n-1)! \) fits:
- For \( n=1 \): \( 1! + 0! = 1 + 1 = 2 \) (correction needed)
- For \( n=2 \): \( 2! + 1! = 2 + 1 = 3 \) (not fitting)
- Possible \( n(n-1) \) seems closer that is n(n-1)/2 seems not rational.
Alternatively, if we rewrite them in terms of multiplication and adding constants: 1, 4, 12, 24 seem like almost factorial related and could fit into polynomial based equations.
Hence, upon inspecting these terms closely and calculating them: The relationship captured seems to revolve in \( 1, 4(n^2) = n(n+1)/2 = 3(n)= 5 \) or \( n^3 \) based outcomes.
Final suggestion seems to state based on factorials proportional to sequence embedded by \( n! \).
Conclusively, the sequence can be best described as having a quasi-factorial pattern. The specific nature can more accurately categorized as some like based on overlaps of combinatorial propagation here \( (n-1)!+(n)!+ \) suggests fantasy output of based functions from zeroth order factorial polynomial sequences per design!
Thus, the sequence is neither arithmetic nor simple geometric but appears related to polynomial growth functions possibly fitting forms of combinatorial values expressed as counts of arrangements.
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