Q. Find an explicit formula for the arithmetic sequence12,5,−2,−9,…. Note: the first term should be a(1).a(n)=□
Identify Type: Identify whether the given sequence is geometric or arithmetic. The sequence 12,5,−2,−9,… has a common difference between consecutive terms, so it is an arithmetic sequence.
Find Common Difference: Determine the common difference (d) of the sequence by subtracting any term from the term that follows it. For example, the difference between the second term (5) and the first term (12) is 5−12=−7.
Use Explicit Formula: Use the explicit formula for an arithmetic sequence, which is an=a1+(n−1)d, where a1 is the first term and d is the common difference. For this sequence, the first term, a1, is 12 and the common difference, d, is −7.
Substitute Values: Substitute the values of a1 and d into the formula to write an expression to describe the sequence. The expression for the sequence is an=12+(n−1)(−7).
Simplify Expression: Simplify the expression by distributing the −7 inside the parentheses. This gives us an=12−7n+7.
Combine Like Terms: Combine like terms in the expression to get the final explicit formula. This results in an=19−7n.
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