f the first 4 terms of a geometric sequence are \( \{7,21,63,189\} \), then ormula for the \( n^{\text {th }} \) term in the sequence is

In a geometric sequence, each term is obtained by multiplying the previous term by a constant ratio \( r \). The general formula for the \( n^{\text{th}} \) term of a geometric sequence is: \[ a_n = a_1 \times r^{n-1} \] where: - \( a_1 \) is the first term. - \( r \) is the common ratio. - \( n \) is the term number. Given the first four terms of the sequence: \( \{7, 21, 63, 189\} \), let's determine the common ratio \( r \): \[ r = \frac{21}{7} = 3 \] Now, using the formula for the \( n^{\text{th}} \) term: \[ a_n = 7 \times 3^{n-1} \] **Verification:** - For \( n = 1 \): \( a_1 = 7 \times 3^{0} = 7 \) - For \( n = 2 \): \( a_2 = 7 \times 3^{1} = 21 \) - For \( n = 3 \): \( a_3 = 7 \times 3^{2} = 63 \) - For \( n = 4 \): \( a_4 = 7 \times 3^{3} = 189 \) All terms match the given sequence. **Final Formula:** \[ a_n = 7 \times 3^{n-1} \] **Answer:** After simplifying, the nth term is 7 multiplied by threeⁿ⁻¹. Thus, aₙ = 7 · 3^{ n−1 }