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Write an explicit formula that represents the sequence defined by the following recursive formula: a_(1)=-7" and "a_(n)=a_(n-1)-5 Answer: a_(n)=

Write an explicit formula that represents the sequence defined by the following recursive formula:\newlinea1=−7 and an=an−1−5 a_{1}=-7 \text { and } a_{n}=a_{n-1}-5 \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=−7 and an=an−1−5 a_{1}=-7 \text { and } a_{n}=a_{n-1}-5 \newlineAnswer: an= a_{n}=
  1. Identify Pattern: The recursive formula given is an=an−1−5a_{n}=a_{n-1}-5, with the initial condition a1=−7a_{1}=-7. To find the explicit formula, we need to determine the pattern of the sequence by looking at the first few terms.
  2. Calculate First Few Terms: Let's calculate the first few terms using the recursive formula:\newlinea2=a1−5=−7−5=−12a_{2} = a_{1} - 5 = -7 - 5 = -12\newlinea3=a2−5=−12−5=−17a_{3} = a_{2} - 5 = -12 - 5 = -17\newlinea4=a3−5=−17−5=−22a_{4} = a_{3} - 5 = -17 - 5 = -22\newlineFrom these calculations, we can see that each term is 55 less than the previous term, which suggests that the sequence is arithmetic with a common difference of −5-5.
  3. Use Arithmetic Sequence Formula: The nnth term of an arithmetic sequence can be found using the formula an=a1+(n−1)da_n = a_1 + (n - 1)d, where a1a_1 is the first term and dd is the common difference. In this case, a1=−7a_1 = -7 and d=−5d = -5.
  4. Substitute Values: Substituting the values of a1a_{1} and dd into the formula, we get:\newlinean=−7+(n−1)(−5)a_{n} = -7 + (n - 1)(-5)\newlinean=−7−5n+5a_{n} = -7 - 5n + 5\newlinean=−5n−2a_{n} = -5n - 2\newlineThis is the explicit formula for the given recursive sequence.

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