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Find an explicit formula for the arithmetic sequence

12,5,-2,-9,dots". "
Note: the first term should be 
a(1).

a(n)=◻_(++_)

Find an explicit formula for the arithmetic sequence\newline12,5,2,9, 12,5,-2,-9, \ldots \text {. } \newlineNote: the first term should be a(1) a(1) .\newlinea(n)= a(n)=\square

Full solution

Q. Find an explicit formula for the arithmetic sequence\newline12,5,2,9, 12,5,-2,-9, \ldots \text {. } \newlineNote: the first term should be a(1) a(1) .\newlinea(n)= a(n)=\square
  1. Identify Type: Identify whether the given sequence is geometric or arithmetic. The sequence 12,5,2,9,12, 5, -2, -9, \ldots has a common difference between consecutive terms, so it is an arithmetic sequence.
  2. Find Common Difference: Determine the common difference (dd) of the sequence by subtracting any term from the term that follows it. For example, the difference between the second term (55) and the first term (1212) is 512=75 - 12 = -7.
  3. Use Explicit Formula: Use the explicit formula for an arithmetic sequence, which is an=a1+(n1)da_n = a_1 + (n-1)d, where a1a_1 is the first term and dd is the common difference. For this sequence, the first term, a1a_1, is 1212 and the common difference, dd, is 7-7.
  4. Substitute Values: Substitute the values of a1a_{1} and dd into the formula to write an expression to describe the sequence. The expression for the sequence is an=12+(n1)(7)a_{n} = 12 + (n-1)(-7).
  5. Simplify Expression: Simplify the expression by distributing the 7-7 inside the parentheses. This gives us an=127n+7a_{n} = 12 - 7n + 7.
  6. Combine Like Terms: Combine like terms in the expression to get the final explicit formula. This results in an=197na_n = 19 - 7n.

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