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Write an explicit formula that represents the sequence defined by the following recursive formula: a_(1)=-4" and "a_(n)=a_(n-1)+2 Answer: a_(n)=

Write an explicit formula that represents the sequence defined by the following recursive formula:\newlinea1=−4 and an=an−1+2 a_{1}=-4 \text { and } a_{n}=a_{n-1}+2 \newlineAnswer: an= a_{n}=

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Q. Write an explicit formula that represents the sequence defined by the following recursive formula:\newlinea1=−4 and an=an−1+2 a_{1}=-4 \text { and } a_{n}=a_{n-1}+2 \newlineAnswer: an= a_{n}=
  1. Identify Pattern in Sequence: Let's find the first few terms of the sequence to identify a pattern:\newlinea1=−4a_{1} = -4\newlinea2=a1+2=−4+2=−2a_{2} = a_{1} + 2 = -4 + 2 = -2\newlinea3=a2+2=−2+2=0a_{3} = a_{2} + 2 = -2 + 2 = 0\newlinea4=a3+2=0+2=2a_{4} = a_{3} + 2 = 0 + 2 = 2\newlineWe can see that each term is increasing by 22 from the previous term.
  2. Find Pattern in Terms of n: Now, let's try to find a pattern in terms of nn:a1=−4=−4+2(1−1)a_{1} = -4 = -4 + 2(1 - 1)a2=−2=−4+2(2−1)a_{2} = -2 = -4 + 2(2 - 1)a3=0=−4+2(3−1)a_{3} = 0 = -4 + 2(3 - 1)a4=2=−4+2(4−1)a_{4} = 2 = -4 + 2(4 - 1)It seems that an=−4+2(n−1)a_{n} = -4 + 2(n - 1)
  3. Simplify the Formula: Let's simplify the formula:\newlinean=−4+2(n−1)a_{n} = -4 + 2(n - 1)\newlinean=−4+2n−2a_{n} = -4 + 2n - 2\newlinean=2n−6a_{n} = 2n - 6\newlineThis is the explicit formula for the given recursive sequence.

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