Write an explicit formula for a_(n), the n^("th ") term of the sequence 3,-12,48,dots Answer: a_(n)=

Identify Pattern in Sequence: The first step is to identify the pattern in the sequence. We notice that each term is multiplied by -4 to get the next term. This indicates that the sequence is geometric with a common ratio of -4.Calculate nth Term Formula: The first term of the sequence, a_1, is 3. Since this is a geometric sequence, the nth term is given by the formula a_n = a_1 * r^(n-1), where r is the common ratio.Substitute Values into Formula: Substituting the known values into the formula, we get a_n = 3 * (-4)^(n-1).Check Formula with First Terms: We should check the formula with the first few terms to ensure there are no mistakes. For n=1, a_n = 3 * (-4)^(1-1) = 3 * 1 = 3, which matches the first term of the sequence. For n=2, a_n = 3 * (-4)^(2-1) = 3 * (-4) = -12, which matches the second term of the sequence. For n=3, a_n = 3 * (-4)^(3-1) = 3 * 16 = 48, which matches the third term of the sequence. The formula appears to be correct.

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Write an explicit formula for
a_(n), the
n^("th ") term of the sequence
3,-12,48,dots
Answer:
a_(n)=

Write an explicit formula for $a_{n}$ , the $n^{\text {th }}$ term of the sequence $3,-12,48, \ldots$ $\newline$ Answer: $a_{n}=$

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Algebra 2

Convert between explicit and recursive formulas

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Q. Write an explicit formula for

a_{n}

, the

n^{\text {th }}

term of the sequence

3,-12,48, \ldots

\newline

Answer:

a_{n}=

Identify Pattern in Sequence: The first step is to identify the pattern in the sequence. We notice that each term is multiplied by $-4$ to get the next term. This indicates that the sequence is geometric with a common ratio of $-4$ .
Calculate nth Term Formula: The first term of the sequence, $a_1$ , is $3$ . Since this is a geometric sequence, the nth term is given by the formula $a_n = a_1 \times r^{(n-1)}$ , where $r$ is the common ratio.
Substitute Values into Formula: Substituting the known values into the formula, we get $a_n = 3 \times (-4)^{(n-1)}$ .
Check Formula with First Terms: We should check the formula with the first few terms to ensure there are no mistakes. For $n=1$ , $a_n = 3 \times (-4)^{1-1} = 3 \times 1 = 3$ , which matches the first term of the sequence. For $n=2$ , $a_n = 3 \times (-4)^{2-1} = 3 \times (-4) = -12$ , which matches the second term of the sequence. For $n=3$ , $a_n = 3 \times (-4)^{3-1} = 3 \times 16 = 48$ , which matches the third term of the sequence. The formula appears to be correct.