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Find an explicit formula for the arithmetic sequence\newline10,10,30,50,.10,-10,-30,-50,\dots.\newlineNote: the first term should be c(1).c(1).\newlinec(n)=c(n)=\square

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Q. Find an explicit formula for the arithmetic sequence\newline10,10,30,50,.10,-10,-30,-50,\dots.\newlineNote: the first term should be c(1).c(1).\newlinec(n)=c(n)=\square
  1. Determine Common Difference: To find an explicit formula for the arithmetic sequence, we first need to determine the common difference dd between the terms. We can do this by subtracting any term from the term that follows it.\newlineSubtract the first term from the second term: 1010=20-10 - 10 = -20.
  2. Write Explicit Formula: Now that we have the common difference, we can write the explicit formula for the arithmetic sequence. The general form of an arithmetic sequence is c(n)=c(1)+(n1)dc(n) = c(1) + (n - 1)d, where c(1)c(1) is the first term and dd is the common difference.\newlineSince the first term c(1)c(1) is 1010 and the common difference dd is 20-20, we can substitute these values into the formula.\newlinec(n)=10+(n1)(20)c(n) = 10 + (n - 1)(-20)
  3. Simplify Formula: Simplify the formula by distributing the common difference 20-20 through the parentheses.\newlinec(n)=1020(n1)c(n) = 10 - 20(n - 1)\newlinec(n)=1020n+20c(n) = 10 - 20n + 20
  4. Combine Like Terms: Combine like terms to get the final explicit formula for the arithmetic sequence.\newlinec(n)=3020nc(n) = 30 - 20n

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