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Find an explicit formula for the geometric sequence -8,-40,-200,-1000,dots.. Note: the first term should be c(1). c(n)=

Find an explicit formula for the geometric sequence −8,−40,−200,−1000,…-8,-40,-200,-1000,\ldots. Note: the first term should be c(1)c(1).\newlinec(n)=c(n)=

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Q. Find an explicit formula for the geometric sequence −8,−40,−200,−1000,…-8,-40,-200,-1000,\ldots. Note: the first term should be c(1)c(1).\newlinec(n)=c(n)=
  1. Identify pattern and type: Identify the pattern in the sequence and determine if it is arithmetic or geometric.\newlineThe given sequence is −8,−40,−200,−1000,…-8, -40, -200, -1000, \ldots\newlineTo determine if the sequence is arithmetic or geometric, we look at the ratio of consecutive terms.\newline−40−8=5\frac{-40}{-8} = 5\newline−200−40=5\frac{-200}{-40} = 5\newline−1000−200=5\frac{-1000}{-200} = 5\newlineSince each term is multiplied by the same number to get the next term, the sequence is geometric.
  2. Find first term and common ratio: Find the first term c(1)c(1) and the common ratio rr of the sequence.\newlineThe first term c(1)c(1) is given as −8-8.\newlineTo find the common ratio, we divide the second term by the first term:\newliner=−40−8=5r = \frac{-40}{-8} = 5
  3. Write explicit formula: Write the explicit formula for the geometric sequence using c(1)c(1) and rr.\newlineThe explicit formula for a geometric sequence is given by:\newlinec(n)=c(1)⋅r(n−1)c(n) = c(1) \cdot r^{(n - 1)}\newlineSubstitute c(1)=−8c(1) = -8 and r=5r = 5 into the formula:\newlinec(n)=−8⋅5(n−1)c(n) = -8 \cdot 5^{(n - 1)}

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