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Complete the recursive formula of the geometric sequence {:[-0.1","-0.5","-2.5","-12.5","dots.],[c(1)=],[c(n)=c(n-1).]:}

Complete the recursive formula of the geometric sequence\newline−0.1,−0.5,−2.5,−12.5,….c(1)=□c(n)=c(n−1)⋅□ \begin{array}{l} -0.1,-0.5,-2.5,-12.5, \ldots . \\ c(1)=\square \\ c(n)=c(n-1) \cdot \square \end{array}

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Q. Complete the recursive formula of the geometric sequence\newline−0.1,−0.5,−2.5,−12.5,….c(1)=□c(n)=c(n−1)⋅□ \begin{array}{l} -0.1,-0.5,-2.5,-12.5, \ldots . \\ c(1)=\square \\ c(n)=c(n-1) \cdot \square \end{array}
  1. Find Common Ratio: We are given the sequence: −0.1,−0.5,−2.5,−12.5,…-0.1, -0.5, -2.5, -12.5, \ldots\newlineTo find the recursive formula, we need to determine the common ratio (r)(r) by dividing any term by the previous term.\newlineLet's divide the second term by the first term to find rr.\newliner=−0.5−0.1=5r = \frac{-0.5}{-0.1} = 5
  2. Write Recursive Formula: Now that we have the common ratio, we can write the recursive formula.\newlineThe recursive formula for a geometric sequence is given by:\newlinec(n)=c(n−1)×rc(n) = c(n-1) \times r\newlineWe already know that r=5r = 5, so we can substitute this value into the formula.\newlinec(n)=c(n−1)×5c(n) = c(n-1) \times 5
  3. Provide First Term: We also need to provide the first term of the sequence for the recursive formula to be complete.\newlineThe first term, c(1)c(1), is given as −0.1-0.1.\newlineSo, the recursive formula is:\newlinec(1)=−0.1c(1) = -0.1\newlinec(n)=c(n−1)×5c(n) = c(n-1) \times 5 for n>1n > 1

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