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Math  /  Algebra

QuestionWrite the explicit rule for the geometric sequence 3,15,75,375,18753,15,75,375,1875.

Studdy Solution

STEP 1

1. The sequence provided is geometric, meaning each term is a constant multiple of the previous term.
2. We need to find the common ratio and the first term to write the explicit rule.

STEP 2

1. Identify the first term of the sequence.
2. Determine the common ratio of the sequence.
3. Write the explicit formula for the geometric sequence.

STEP 3

Identify the first term of the sequence. The first term a1 a_1 is:
a1=3 a_1 = 3

STEP 4

Determine the common ratio r r by dividing the second term by the first term:
r=153=5 r = \frac{15}{3} = 5
Verify the common ratio by checking other consecutive terms:
7515=5,37575=5,1875375=5 \frac{75}{15} = 5, \quad \frac{375}{75} = 5, \quad \frac{1875}{375} = 5
The common ratio is consistent throughout the sequence.

STEP 5

Write the explicit formula for the geometric sequence. The formula for the n n -th term of a geometric sequence is given by:
an=a1⋅rn−1 a_n = a_1 \cdot r^{n-1}
Substitute the values of a1 a_1 and r r :
an=3⋅5n−1 a_n = 3 \cdot 5^{n-1}
This is the explicit rule for the given geometric sequence.

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