$$ ext { 41) Write an explicit formula for the recursive formula. } \ a_{1}=16, a_{n}=4 a_{n-1}, n \geq 2 \ a_{1} \ \ a_{n}= $$

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$$ 	ext { 41) Write an explicit formula for the recursive formula. } \ a_{1}=16, a_{n}=4 a_{n-1}, n \geq 2 \ a_{1} \ \ a_{n}= $$

UpStudy AI · Accepted Answer

To find an explicit formula for the recursive sequence defined by:

- $$ a_1 = 16 $$
- $$ a_n = 4 a_{n-1} $$ for $$ n \geq 2 $$

we can analyze the recursive relationship step by step.

1. **Identify the pattern**:
   - For $$ n = 2 $$:
     $$
     a_2 = 4 a_1 = 4 \times 16 = 64
     $$
   - For $$ n = 3 $$:
     $$
     a_3 = 4 a_2 = 4 \times 64 = 256
     $$
   - For $$ n = 4 $$:
     $$
     a_4 = 4 a_3 = 4 \times 256 = 1024
     $$

2. **Generalize the pattern**:
   - We can see that each term is obtained by multiplying the previous term by 4. This suggests that the sequence is a geometric sequence where the first term $$ a_1 = 16 $$ and the common ratio $$ r = 4 $$.

3. **Explicit formula**:
   - The explicit formula for a geometric sequence can be expressed as:
     $$
     a_n = a_1 \cdot r^{n-1}
     $$
   - Substituting the known values:
     $$
     a_n = 16 \cdot 4^{n-1}
     $$

Thus, the explicit formula for the recursive sequence is:

$$
a_n = 16 \cdot 4^{n-1}
$$
The explicit formula for the sequence is $$ a_n = 16 \times 4^{n-1} $$.

\( \text { 41) Write an explicit formula for the recursive formula. } \\ a_{1}=16, a_{n}=4 a_{n-1}, n \geq 2 \\ a_{1} \ \ a_{n}= \)

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