QuestionAugust 24, 2025

Write an equation to describe the sequence below. Use n to represent the position of a term in the sequence , where n = 1 for the first term. 3,12,48,ldots Write your answer using decimals and integers. a_(n)=square (square )^n-1

Write an equation to describe the sequence below. Use n to represent the position of a term in the sequence , where n = 1 for the first term. 3,12,48,ldots Write your answer using decimals and integers. a_(n)=square (square )^n-1
Write an equation to describe the sequence below. Use n to represent the position of a term in the sequence , where n = 1 for the first term. 3,12,48,ldots Write your answer using decimals and integers. a_(n)=square (square )^n-1

Solution
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Answer

a_{n} = 3(4)^{n-1} Explanation 1. Identify the Pattern The sequence is geometric. Each term is multiplied by 4 to get the next term. 2. Determine the First Term and Common Ratio The first term a_1 = 3. The common ratio r = \frac{12}{3} = 4. 3. Write the General Formula for a Geometric Sequence **a_n = a_1 \cdot r^{n-1}** 4. Substitute Values into the Formula Substitute a_1 = 3 and r = 4 into the formula: a_n = 3 \cdot 4^{n-1}.

Explanation

1. Identify the Pattern<br /> The sequence is geometric. Each term is multiplied by 4 to get the next term.<br /><br />2. Determine the First Term and Common Ratio<br /> The first term $a_1 = 3$. The common ratio $r = \frac{12}{3} = 4$.<br /><br />3. Write the General Formula for a Geometric Sequence<br /> **$a_n = a_1 \cdot r^{n-1}$**<br /><br />4. Substitute Values into the Formula<br /> Substitute $a_1 = 3$ and $r = 4$ into the formula: $a_n = 3 \cdot 4^{n-1}$.
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