Convert the recursive form a1=5, an=2an−1a1=5, an=2an−1 to the explicit form. (1 point) Responses an=3(7)nan=3(7)nA sub n is equal to 3 times 7 to the n th power an=5(2)n+1an=5(2)n+1A sub n is equal to 5 times 2 raised to the n plus 1 power an=5(2)n−1an=5(2)n−1A sub n is equal to 5 times 2 raised to the n minus 1 power an=2(5)n−1

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GPT-4o mini · Answer

To convert the recursive form $ a_1 = 5 $ and $ a_n = 2 a_{n-1} $ into an explicit form, we can calculate the first few terms: - $ a_1 = 5 $ - $ a_2 = 2 a_1 = 2 \cdot 5 = 10 $ - $ a_3 = 2 a_2 = 2 \cdot 10 = 20 $ - $ a_4 = 2 a_3 = 2 \cdot 20 = 40 $ It appears that the terms are multiplying by 2 each time. Let's notice the pattern: - $ a_1 = 5 $ - $ a_2 = 5 \cdot 2^1 $ - $ a_3 = 5 \cdot 2^2 $ - $ a_4 = 5 \cdot 2^3 $ From this, we can conclude that in general: $$ a_n = 5 \cdot 2^{n-1} $$ So the explicit form of the sequence is: $$ a_n = 5 \cdot 2^{n-1} $$ Therefore, the correct response is: $$ a_n = 5(2)^{n-1} $$