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Math  /  Algebra

QuestionFind a formula for the general term ana_{n} of the sequence assuming the pattern of the first few terms continues. {72,74,78,716,732,…}\left\{\frac{7}{2}, \frac{7}{4}, \frac{7}{8}, \frac{7}{16}, \frac{7}{32}, \ldots\right\}
Assume the first term is a1a_{1}. an=a_{n}=

Studdy Solution

STEP 1

1. The sequence is geometric.
2. The first term of the sequence is a1=72 a_1 = \frac{7}{2} .
3. The pattern of the sequence continues as observed.

STEP 2

1. Identify the common ratio of the sequence.
2. Write the general formula for the n n -th term of a geometric sequence.
3. Substitute the identified values into the general formula.

STEP 3

To find the common ratio r r , divide the second term by the first term: r=7472=74×27=12 r = \frac{\frac{7}{4}}{\frac{7}{2}} = \frac{7}{4} \times \frac{2}{7} = \frac{1}{2}

STEP 4

The general formula for the n n -th term of a geometric sequence is: an=a1⋅rn−1 a_n = a_1 \cdot r^{n-1}

STEP 5

Substitute a1=72 a_1 = \frac{7}{2} and r=12 r = \frac{1}{2} into the formula: an=72⋅(12)n−1 a_n = \frac{7}{2} \cdot \left(\frac{1}{2}\right)^{n-1}
The formula for the general term an a_n of the sequence is:
an=72n a_n = \frac{7}{2^n}

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