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Write an explicit formula that represents the sequence defined by the following recursive formula: a_(1)=144" and "a_(n)=(1)/(6)a_(n-1) Answer: a_(n)=

Write an explicit formula that represents the sequence defined by the following recursive formula:\newlinea1=144 and an=16an−1 a_{1}=144 \text { and } a_{n}=\frac{1}{6} a_{n-1} \newlineAnswer: an= a_{n}=

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Q. Write an explicit formula that represents the sequence defined by the following recursive formula:\newlinea1=144 and an=16an−1 a_{1}=144 \text { and } a_{n}=\frac{1}{6} a_{n-1} \newlineAnswer: an= a_{n}=
  1. Identify first term and formula: Identify the first term and the recursive formula.\newlineThe first term is given as a1=144a_{1} = 144. The recursive formula is given as an=16an−1a_{n} = \frac{1}{6}a_{n-1}, which means each term is one-sixth of the previous term.
  2. Recognize sequence type: Recognize the type of sequence. Since each term is a constant fraction of the previous term, this is a geometric sequence.
  3. Determine common ratio: Determine the common ratio rr of the geometric sequence.\newlineThe common ratio rr is the factor that each term is multiplied by to get the next term. From the recursive formula, r=16r = \frac{1}{6}.
  4. Write explicit formula: Write the explicit formula for a geometric sequence.\newlineThe explicit formula for a geometric sequence is an=a1×r(n−1)a_{n} = a_{1} \times r^{(n-1)}, where a1a_{1} is the first term and rr is the common ratio.
  5. Substitute values into formula: Substitute the values of a1a_{1} and rr into the explicit formula.\newlinea1=144a_{1} = 144 and r=16r = \frac{1}{6}, so the explicit formula becomes an=144×(16)n−1a_{n} = 144 \times \left(\frac{1}{6}\right)^{n-1}.

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