Find the explicit formula that produces the given sequence:
\[ \frac{3}{2}, \frac{3}{4}, \frac{3}{8}, \frac{3}{16}, \ldots \]
Answer (2)
2 3 ; 4 3 ; 8 3 ; 16 3 ; ... a 1 = 2 3 a 2 = 4 3 = 2 3 ⋅ 2 1 a 3 = 8 3 = 4 3 ⋅ 2 1 = 2 3 ⋅ ( 2 1 ) 2 a 4 = 16 3 = 8 3 ⋅ 2 1 = 2 3 ⋅ ( 2 1 ) 3 ⋮ a n = 2 3 ⋅ ( 2 1 ) n − 1 = 2 3 ⋅ ( 2 1 ) − 1 ⋅ ( 2 1 ) n = 2 3 ⋅ 2 ⋅ ( 2 1 ) n = 3 ⋅ ( 2 1 ) n
The explicit formula for the sequence 2 3 , 4 3 , 8 3 , 16 3 , … is a n = 2 n 3 . This formula allows us to find any term in the sequence by plugging in the value of n. The sequence represents a geometric pattern where each term is half of the one before it, multiplied by 3 in the numerator.
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