State an explicit rule for the nth term of the recursively-defined sequence.
-$$a _ { n } = \left( \frac { 1 } { 7 } \right) \cdot a _ { n - 1 } ; a _ { 1 } = 1$$
A) $a _ { n } = 1 \cdot \left( \frac { 1 } { 7 } \right) ^ { n }$
B) $a _ { n } = 7 \cdot \left( \frac { 1 } { 1 } \right) ^ { n }$
C) $ a _ { n } = 7 \cdot \left( \frac { 1 } { 1 } \right) ^ { n - 1 }$
D) $a _ { n } = 1 \cdot \left( \frac { 1 } { 7 } \right) ^ { n - 1 }$

Question

Quizplus · Accepted Answer

The rule for the nth term of the sequence is derived by applying the recursive formula repeatedly. Starting with $a_1 = 1$, each subsequent term is obtained by multiplying the previous term by $\frac{1}{7}$. Thus, the nth term is $1$ (the initial value) times $\left(\frac{1}{7}\right)$ raised to the power of $n-1$, because we multiply by $\frac{1}{7}$ for each step from the first term to the nth term, which is $n-1$ steps.

State an Explicit Rule for the Nth Term of the Recursively-Defined