To find a formula for the sequence: 2, 3.5, 6.125, 10.71875..., we can start by examining the differences or ratios between the terms.
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Identify the terms:
\( a_1 = 2 \)
\( a_2 = 3.5 \)
\( a_3 = 6.125 \)
\( a_4 = 10.71875 \) -
Examine the ratios:
To discover how the terms relate to each other, we can try to find a pattern using ratios:
- \( \frac{a_2}{a_1} = \frac{3.5}{2} = 1.75 \)
- \( \frac{a_3}{a_2} = \frac{6.125}{3.5} \approx 1.75 \)
- \( \frac{a_4}{a_3} = \frac{10.71875}{6.125} \approx 1.75 \)
Observing that the ratio of each term to the previous term is approximately 1.75 suggests that the sequence may be geometric or have a geometric progression-type formula.
- Hypothesizing a General Formula:
Given this ratio of approximately 1.75, we can express the terms of the sequence in terms of the first term multiplied by a common ratio raised to increasing powers:
- \( a_n = 2 \cdot (1.75)^{n-1} \)
- Verification:
Let’s calculate the first four terms using the proposed formula:
- For \( n=1 \):
\( a_1 = 2 \cdot (1.75)^{0} = 2 \) - For \( n=2 \):
\( a_2 = 2 \cdot (1.75)^{1} = 2 \cdot 1.75 = 3.5 \) - For \( n=3 \):
\( a_3 = 2 \cdot (1.75)^{2} = 2 \cdot 3.0625 = 6.125 \) - For \( n=4 \):
\( a_4 = 2 \cdot (1.75)^{3} \approx 2 \cdot 5.3789 \approx 10.71875 \)
Since the computed values match the sequence given, we conclude that: The formula for the sequence is:
\[ a_n = 2 \cdot (1.75)^{n-1} \]
This accurately represents the series.