Which formula can be used to describe the sequence -2 2/3, -5 1/3, -10 2/3, -21 1/3, -42 2/3?
Answer
The sequence can be defined by the recursive formula f(n)=2 x f(n-1) with f(0)=-8/3 or by the closed-form formula f(n)=-(8/3) x 2^n. This indicates that each subsequent term is double the previous term, starting from -2 2/3. Both representations capture the growth of the sequence accurately.
Explanation
To understand the sequence -2 2/3, -5 1/3, -10 2/3, -21 1/3, -42 2/3, we need to look for a pattern or a formula that defines it. 1. Convert the Mixed Numbers to Improper Fractions: -2 2/3 = -8/3 -5 1/3 = -16/3 -10 2/3 = -32/3 -21 1/3 = -64/3 -42 2/3 = -128/3 This gives us the sequence: -8/3, -16/3, -32/3, -64/3, -128/3. 2. Identify the Pattern: Each term is obtained by multiplying the previous term by 2, identifying this as a geometric sequence with a common ratio r = 2. -16/3 = 2 x (-8/3) -32/3 = 2 x (-16/3) 3. General Formulas: - Recursive Formula: f(n) = 2 x f(n-1) with initial term f(0) = -8/3. - Closed Form (Explicit): f(n) = -(8/3) x 2^n, where n represents the number of doublings from the first term (n=0, 1, 2...). This confirms that each term doubles the previous value while maintaining the negative trajectory.