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Find an explicit formula for the geometric sequence 2\,,\,6\,,\,18\,,\,54,\unicode{0x2026}. Note: the first term should be b(1)b(1).

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Q. Find an explicit formula for the geometric sequence 2\,,\,6\,,\,18\,,\,54,\unicode{0x2026}. Note: the first term should be b(1)b(1).
  1. Identify Terms and Ratio: We need to identify the first term and the common ratio of the geometric sequence. The first term is given as 22, which will be b(1)b(1) in our formula. To find the common ratio, we divide the second term by the first term, the third term by the second term, and so on, to ensure they are all the same.\newlineCalculation: \newliner=62=3r = \frac{6}{2} = 3\newliner=186=3r = \frac{18}{6} = 3\newliner=5418=3r = \frac{54}{18} = 3
  2. Calculate Common Ratio: Now that we have the first term b(1)=2b(1) = 2 and the common ratio r=3r = 3, we can write the explicit formula for the nth term of a geometric sequence, which is b(n)=b(1)×r(n−1)b(n) = b(1) \times r^{(n-1)}.\newlineCalculation: \newlineb(n)=2×3(n−1)b(n) = 2 \times 3^{(n-1)}

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