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{[b(1)=-54],[b(n)=b(n-1)*(4)/(3)]:} What is the 4^("th ") term in the sequence?

{b(1)=−54b(n)=b(n−1)⋅43 \left\{\begin{array}{l} b(1)=-54 \\ b(n)=b(n-1) \cdot \frac{4}{3} \end{array}\right. \newlineWhat is the 4th  4^{\text {th }} term in the sequence?

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Full solution

Q. {b(1)=−54b(n)=b(n−1)⋅43 \left\{\begin{array}{l} b(1)=-54 \\ b(n)=b(n-1) \cdot \frac{4}{3} \end{array}\right. \newlineWhat is the 4th  4^{\text {th }} term in the sequence?
  1. Identify sequence type: Identify the type of sequence.\newlineThe sequence is defined recursively with each term being a multiple of the previous term. This indicates that the sequence is geometric.
  2. Determine common ratio: Determine the common ratio of the sequence.\newlineThe common ratio is given by the recursive formula b(n)=b(n−1)×(43)b(n) = b(n-1) \times \left(\frac{4}{3}\right). Therefore, the common ratio is 43\frac{4}{3}.
  3. Calculate second term: Calculate the second term using the first term and the common ratio.\newlineb(2)=b(1)×(43)=−54×(43)=−72b(2) = b(1) \times \left(\frac{4}{3}\right) = -54 \times \left(\frac{4}{3}\right) = -72
  4. Calculate third term: Calculate the third term using the second term and the common ratio.\newlineb(3)=b(2)×(43)=−72×(43)=−96b(3) = b(2) \times \left(\frac{4}{3}\right) = -72 \times \left(\frac{4}{3}\right) = -96
  5. Calculate fourth term: Calculate the fourth term using the third term and the common ratio.\newlineb(4)=b(3)×(43)=−96×(43)=−128b(4) = b(3) \times \left(\frac{4}{3}\right) = -96 \times \left(\frac{4}{3}\right) = -128

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