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Write an explicit formula that represents the sequence defined by the following recursive formula: a_(1)=3" and "a_(n)=2a_(n-1) Answer: a_(n)=

Write an explicit formula that represents the sequence defined by the following recursive formula:\newlinea1=3 and an=2an−1 a_{1}=3 \text { and } a_{n}=2 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=3 and an=2an−1 a_{1}=3 \text { and } a_{n}=2 a_{n-1} \newlineAnswer: an= a_{n}=
  1. Identify First Term: Identify the first term and the recursive pattern of the sequence.\newlineThe first term is given as a1=3a_{1}=3. The recursive pattern is that each term is twice the previous term, which is given by an=2an−1a_{n}=2a_{n-1}.
  2. Determine Explicit Formula: Determine the explicit formula based on the recursive pattern.\newlineSince each term is twice the previous term, we can write the first few terms to see the pattern:\newlinea1=3a_{1} = 3\newlinea2=2a1=2×3=6a_{2} = 2a_{1} = 2\times3 = 6\newlinea3=2a2=2×6=12a_{3} = 2a_{2} = 2\times6 = 12\newlinea4=2a3=2×12=24a_{4} = 2a_{3} = 2\times12 = 24\newline...\newlineWe can see that each term is 22 raised to the power of (n−1)(n-1) times the first term (3)(3).
  3. Write Formula: Write the explicit formula using the pattern observed.\newlineThe explicit formula is an=3×2(n−1)a_{n} = 3 \times 2^{(n-1)}.

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