Question

understanding geometric sequences

suppose a term of a geometric sequence is \\(a_4 = 121.5\\) and the common ratio is 3. write the formula for this sequence in the form \\(a_n = a_1 \cdot r^{n-1}\\). explain how you arrived at your answer.

Explanation:

Find the first term of the sequence

Using the Geometric Sequence Explicit Formula and Common Ratio knowledge points

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Write the explicit formula

Using the Geometric Sequence Explicit Formula knowledge point

$$ a_n = 4.5 \cdot 3^{n-1} $$

Answer:

To find the formula for the geometric sequence, we use the explicit form \(a_n = a_1 \cdot r^{n-1}\).

1. Substitute the given values \(a_4 = 121.5\) and \(r = 3\) into the formula to solve for the first term \(a_1\):

$$ 121.5 = a_1 \cdot 3^{4-1} $$

$$ 121.5 = a_1 \cdot 27 $$

$$ a_1 = \frac{121.5}{27} = 4.5 $$

1. Substitute \(a_1 = 4.5\) and \(r = 3\) back into the general form to write the final sequence formula:

$$ a_n = 4.5 \cdot 3^{n-1} $$