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Write an explicit formula that represents the sequence defined by the following recursive formula: a_(1)=50" and "a_(n)=(1)/(5)a_(n-1) Answer: a_(n)=

Write an explicit formula that represents the sequence defined by the following recursive formula:\newlinea1=50 and an=15an−1 a_{1}=50 \text { and } a_{n}=\frac{1}{5} 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=50 and an=15an−1 a_{1}=50 \text { and } a_{n}=\frac{1}{5} a_{n-1} \newlineAnswer: an= a_{n}=
  1. Identify First Term: Identify the first term of the sequence.\newlineThe first term a1a_{1} is given as 5050.
  2. Recognize Recursive Pattern: Recognize the pattern of the recursive formula.\newlineThe recursive formula an=15an−1a_n=\frac{1}{5}a_{n-1} indicates that each term is 15\frac{1}{5} times the previous term.
  3. Determine Common Ratio: Determine the common ratio of the geometric sequence. Since each term is (15)(\frac{1}{5}) times the previous term, the common ratio rr is (15)(\frac{1}{5}).
  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: Substitute the values of a1a_{1} and rr into the explicit formula.\newlinea1=50a_{1} = 50 and r=15r = \frac{1}{5}, so the explicit formula becomes an=50×(15)n−1a_{n} = 50 \times \left(\frac{1}{5}\right)^{n-1}.

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