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

Write an explicit formula that represents the sequence defined by the following recursive formula:\newlinea1=−5 and an=an−1+3 a_{1}=-5 \text { and } a_{n}=a_{n-1}+3 \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=an−1+3 a_{1}=-5 \text { and } a_{n}=a_{n-1}+3 \newlineAnswer: an= a_{n}=
  1. Initial Term and Recursive Formula: The first term of the sequence is given as a1=−5a_{1} = -5. To move from one term to the next, we add 33 each time according to the recursive formula an=an−1+3a_{n} = a_{n-1} + 3.
  2. Finding Next Terms: Let's find the next few terms to identify a pattern:\newlinea2=a1+3=−5+3=−2a_{2} = a_{1} + 3 = -5 + 3 = -2\newlinea3=a2+3=−2+3=1a_{3} = a_{2} + 3 = -2 + 3 = 1\newlinea4=a3+3=1+3=4a_{4} = a_{3} + 3 = 1 + 3 = 4\newlineWe can see that each term is 33 more than the previous term.
  3. Expressing nth Term: Now, let's express the nnth term in terms of the first term and the common difference (which is 33):an=a1+(n−1)×3a_{n} = a_{1} + (n - 1) \times 3We know a1=−5a_{1} = -5, so we substitute that into the equation:an=−5+(n−1)×3a_{n} = -5 + (n - 1) \times 3
  4. Simplifying Equation: Simplify the equation:\newlinean=−5+3n−3a_{n} = -5 + 3n - 3\newlinean=3n−8a_{n} = 3n - 8\newlineThis is the explicit formula for the given recursive sequence.

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