If the first 5 terms of a geometric sequence are \( \left\{12,6,3, \frac{3}{2}, \frac{3}{4}\right\} \), then the formula for the \( n^{\text {th }} \) term in the sequence is

A geometric sequence is defined by its first term \( a \) and a common ratio \( r \) such that each subsequent term is obtained by multiplying the previous term by \( r \). Given the first five terms of the sequence: \[ 12, 6, 3, \frac{3}{2}, \frac{3}{4} \] 1. **Identify the first term (\( a \)):** \[ a = 12 \] 2. **Determine the common ratio (\( r \)):** \[ r = \frac{6}{12} = \frac{1}{2} \] To confirm, check subsequent terms: \[ \frac{3}{6} = \frac{1}{2}, \quad \frac{\frac{3}{2}}{3} = \frac{1}{2}, \quad \frac{\frac{3}{4}}{\frac{3}{2}} = \frac{1}{2} \] So, \( r = \frac{1}{2} \) is consistent. 3. **Formulate the \( n^{\text{th}} \) term:** The general formula for the \( n^{\text{th}} \) term of a geometric sequence is: \[ a_n = a \cdot r^{n-1} \] Substituting the known values: \[ a_n = 12 \left( \frac{1}{2} \right)^{n-1} \] **Final Formula:** \[ a_n = 12 \cdot \left( \frac{1}{2} \right)^{\,n-1} \] Answer: After simplifying, the nth term is 12 multiplied by (½)ⁿ⁻¹. Thus, aₙ = 12 · (1/2)^(n−1)