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

Write an explicit formula that represents the sequence defined by the following recursive formula:\newlinea1=1 and an=−2an−1 a_{1}=1 \text { and } a_{n}=-2 a_{n-1} \newlineAnswer: an= a_{n}=

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Full solution

Q. Write an explicit formula that represents the sequence defined by the following recursive formula:\newlinea1=1 and an=−2an−1 a_{1}=1 \text { and } a_{n}=-2 a_{n-1} \newlineAnswer: an= a_{n}=
  1. Identify Pattern: The recursive formula given is an=−2an−1a_{n} = -2a_{n-1}, with the initial condition a1=1a_{1} = 1. To find the explicit formula, we will look at the first few terms of the sequence to identify a pattern.
  2. Calculate a2a_{2}: First, we calculate a2a_{2} using the recursive formula:\newlinea2=−2a1=−2(1)=−2a_{2} = -2a_{1} = -2(1) = -2.
  3. Calculate a3a_{3}: Next, we calculate a3a_{3}:a3=−2a2=−2(−2)=4a_{3} = -2a_{2} = -2(-2) = 4.
  4. Calculate a4a_{4}: Now, we calculate a4a_{4}:a4=−2a3=−2(4)=−8a_{4} = -2a_{3} = -2(4) = -8.
  5. Geometric Sequence: We can see that each term is −2-2 times the previous term. This is a geometric sequence with the first term a1=1a_{1} = 1 and a common ratio r=−2r = -2. The explicit formula for a geometric sequence is an=a1×r(n−1)a_{n} = a_{1} \times r^{(n-1)}.
  6. Substitute Values: Substituting the values of a1a_{1} and rr into the formula, we get:\newlinean=1×(−2)n−1.a_{n} = 1 \times (-2)^{n-1}.
  7. Final Explicit Formula: Simplifying the formula, we have the explicit formula for the sequence: an=(−2)n−1a_{n} = (-2)^{n-1}.

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