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Write an explicit formula that represents the sequence defined by the following recursive formula: a_(1)=5" and "a_(n)=-3a_(n-1) Answer: a_(n)=

Write an explicit formula that represents the sequence defined by the following recursive formula:\newlinea1=5 and an=−3an−1 a_{1}=5 \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=5 and an=−3an−1 a_{1}=5 \text { and } a_{n}=-3 a_{n-1} \newlineAnswer: an= a_{n}=
  1. Identify Terms: To find the explicit formula, we start by looking at the first few terms of the sequence using the recursive formula. \newlinea1=5a_{1} = 5 (given)\newlinea2=−3a1=−3(5)=−15a_{2} = -3a_{1} = -3(5) = -15\newlinea3=−3a2=−3(−15)=45a_{3} = -3a_{2} = -3(-15) = 45\newlinea4=−3a3=−3(45)=−135a_{4} = -3a_{3} = -3(45) = -135
  2. Recognize Pattern: We notice a pattern where each term is −3-3 times the previous term. This is a geometric sequence with the first term a1=5a_{1} = 5 and a common ratio r=−3r = -3. The explicit formula for a geometric sequence is given by: an=a1×r(n−1)a_{n} = a_{1} \times r^{(n-1)}
  3. Apply Explicit Formula: Substitute the values of a1=5a_{1} = 5 and r=−3r = -3 into the formula to get the explicit formula for this sequence:\newlinean=5⋅(−3)n−1a_{n} = 5 \cdot (-3)^{n-1}

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