The recursive relation is given as:
- \( a_1 = 5 \)
- \( a_n = 2a_{n-1} \) for \( n > 1 \)
To derive the explicit form, let's analyze the terms step by step:
-
For \( n = 1 \): \[ a_1 = 5 \]
-
For \( n = 2 \): \[ a_2 = 2a_1 = 2 \times 5 = 10 \]
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For \( n = 3 \): \[ a_3 = 2a_2 = 2 \times 10 = 20 \]
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For \( n = 4 \): \[ a_4 = 2a_3 = 2 \times 20 = 40 \]
From the above calculations, we can see a pattern forming. We can observe that:
\[ a_n = 2a_{n-1} \]
This suggests each term is generated by multiplying the previous term by 2. We can express any term \( a_n \) as follows:
\[ a_n = 2^{n-1} a_1 \]
Substituting \( a_1 = 5 \):
\[ a_n = 2^{n-1} \times 5 \]
Thus, the explicit form can be written as:
\[ a_n = 5 \cdot 2^{n-1} \]
Therefore, the correct response is:
an=5(2)n−1
A sub n is equal to 5 times 2 raised to the n minus 1 power.
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