[tex]\bf 81~~,~~\stackrel{81-27}{54}~~,~~\stackrel{54-27}{27}~~,~~\stackrel{27-27}{0}~~,~...\qquad \qquad \stackrel{\textit{common difference}}{d = -27} \\\\[-0.35em] ~\dotfill\\\\ n^{th}\textit{ term of an arithmetic sequence} \\\\ a_n=a_1+(n-1)d\qquad \begin{cases} a_n=n^{th}\ term\\ n=\textit{term position}\\ a_1=\textit{first term}\\ d=\textit{common difference}\\ \cline{1-1} a_1=81\\ d=-27 \end{cases} \\\\\\ a_n=81+(n-1)(-27)\implies a_n=81-27n+27\implies a_n=108-27n[/tex]
An arithmetic sequence is a sequence in which the difference between any two consecutive terms of the sequence is equal. The explicit formula for the given arithmetic sequence is aⁿ = 81 + (n-1)(-27).
An arithmetic sequence is a sequence in which the difference between any two consecutive terms of the sequence is equal.
aₙ = a₁ + (n-1)d
where,
aₙ is the nth term of the sequence,
a₁ is the first term of the sequence,
d is a common difference between every two terms.
For an arithmetic sequence, the common difference between any two consecutive terms is the same. Therefore, an explicit formula for the given arithmetic sequence is,
Difference, d = 54 - 81 = -27
Also, the first term of the sequence is 27. Therefore, the explicit formula for the given arithmetic sequence is,
aₙ = a₁ + (n-1)d
aₙ = 81 + (n-1)(-27)
Hence, the explicit formula for the given arithmetic sequence is aⁿ = 81 + (n-1)(-27).
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