Convert the recursive form: a1 = 5, an = 2an-1, to the explicit form

Answer 1

To convert the recursive formula \( a_1 = 5 \) and \( a_n = 2a_{n-1} \) into an explicit form, we can analyze the pattern of the sequence.

We can calculate the first few terms of the sequence:

1. \( a_1 = 5 \)
2. \( a_2 = 2a_1 = 2 \cdot 5 = 10 \)
3. \( a_3 = 2a_2 = 2 \cdot 10 = 20 \)
4. \( a_4 = 2a_3 = 2 \cdot 20 = 40 \)
5. \( a_5 = 2a_4 = 2 \cdot 40 = 80 \)

From these calculations, we can observe that:

• \( a_1 = 5 \)
• \( a_2 = 10 = 5 \cdot 2^1 \)
• \( a_3 = 20 = 5 \cdot 2^2 \)
• \( a_4 = 40 = 5 \cdot 2^3 \)
• \( a_5 = 80 = 5 \cdot 2^4 \)

It appears that the general term \( a_n \) can be expressed as:

\[ a_n = 5 \cdot 2^{n-1} \]

Therefore, the explicit form of the sequence is:

\[ a_n = 5 \cdot 2^{n-1} \]

for \( n \geq 1 \).