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

Write an explicit formula that represents the sequence defined by the following recursive formula:\newlinea1=1 and an=−3an−1 a_{1}=1 \text { and } a_{n}=-3 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=1 and an=−3an−1 a_{1}=1 \text { and } a_{n}=-3 a_{n-1} \newlineAnswer: an= a_{n}=
  1. Identify Pattern: To find the explicit formula for the sequence, we start by looking at the first few terms to identify a pattern.\newlineGiven: a1=1a_{1} = 1\newlineUsing the recursive formula an=−3an−1a_{n} = -3a_{n-1}, we find the next few terms:\newlinea2=−3a1=−3(1)=−3a_{2} = -3a_{1} = -3(1) = -3\newlinea3=−3a2=−3(−3)=9a_{3} = -3a_{2} = -3(-3) = 9\newlinea4=−3a3=−3(9)=−27a_{4} = -3a_{3} = -3(9) = -27\newlineWe can see that each term is −3-3 times the previous term, which suggests a geometric sequence with a common ratio of −3-3.
  2. Explicit Formula: The general form of an explicit formula for a geometric sequence is an=a1⋅r(n−1)a_n = a_1 \cdot r^{(n-1)}, where a1a_1 is the first term and rr is the common ratio.\newlineIn our case, a1=1a_1 = 1 and r=−3r = -3.\newlineSo, the explicit formula is an=1⋅(−3)(n−1)a_n = 1 \cdot (-3)^{(n-1)}.
  3. Simplify Formula: We simplify the explicit formula to get the final answer. \newlinean=(−3)n−1a_{n} = (-3)^{n-1}\newlineThis is the explicit formula for the given recursive sequence.

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