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Complete the recursive formula of the geometric sequence {:[-0.25","-2","-16","-128","dots.],[b(1)=],[b(n)=b(n-1).]:}

Complete the recursive formula of the geometric sequence\newline−0.25,−2,−16,−128,….b(1)=□b(n)=b(n−1)⋅□ \begin{array}{l} -0.25,-2,-16,-128, \ldots . \\ b(1)=\square \\ b(n)=b(n-1) \cdot \square \end{array}

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Q. Complete the recursive formula of the geometric sequence\newline−0.25,−2,−16,−128,….b(1)=□b(n)=b(n−1)⋅□ \begin{array}{l} -0.25,-2,-16,-128, \ldots . \\ b(1)=\square \\ b(n)=b(n-1) \cdot \square \end{array}
  1. Given Sequence: We are given the geometric sequence: −0.25,−2,−16,−128,…-0.25, -2, -16, -128, \ldots To find the recursive formula, we need to determine the common ratio rr by dividing any term by its preceding term. Let's divide the second term by the first term: r=(−2)/(−0.25)=8r = (-2) / (-0.25) = 8.
  2. Common Ratio: Now that we have the common ratio, we can write the recursive formula. The first term b(1)b(1) is given as −0.25-0.25. The recursive formula will relate term b(n)b(n) to the previous term b(n−1)b(n-1) using the common ratio.\newlineThe recursive formula is: b(n)=b(n−1)×rb(n) = b(n-1) \times r, where r=8r = 8.
  3. Recursive Formula: Let's verify the recursive formula by applying it to find the third term using the second term.\newlineAccording to the formula, b(3)b(3) should equal b(2)×rb(2) \times r.\newlineWe have b(2)=−2b(2) = -2 and r=8r = 8, so b(3)=−2×8=−16b(3) = -2 \times 8 = -16, which matches the given sequence.

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