Question

karton · Accepted Answer

The explicit formula for the geometric sequence is given by:

a_{n} = 5 {(- \frac{1}{8})}^{n - 1}

To find the recursive formula for this geometric sequence, we can express

a_{n}

in terms of its preceding term

a_{n - 1}

.
Let's consider the ratio between consecutive terms:

r = \frac{a_{n}}{a_{n - 1}}

Substituting the given explicit formula, we have:

r = \frac{5 {(- \frac{1}{8})}^{n - 1}}{5 {(- \frac{1}{8})}^{(n - 1) - 1}}

Simplifying, we get:

r = - \frac{1}{8}

Therefore, the common ratio

r

between consecutive terms is constant and equal to

- \frac{1}{8}

.
Now, we can express the recursive formula for the geometric sequence:

a_{n} = r \cdot a_{n - 1}

Substituting the value of

r

, we have:

a_{n} = (- \frac{1}{8}) \cdot a_{n - 1}

Hence, the recursive formula for the given geometric sequence with the explicit formula

a_{n} = 5 {(- \frac{1}{8})}^{n - 1}

is:

a_{n} = - \frac{1}{8} \cdot a_{n - 1}

.

Answered question