* 3. \( x ; x+4 ; \ldots \). is a geometric sequence. Determine the third term in terms of \( x \).

To find the third term of a geometric sequence, we need to know the first term and the common ratio. Given that the sequence is \( x, x+4, \ldots \), we can see that the first term is \( x \) and the common ratio is \( \frac{x+4}{x} \). The formula for the nth term of a geometric sequence is: \[ a_n = a_1 \times r^{(n-1)} \] where: - \( a_n \) is the nth term, - \( a_1 \) is the first term, - \( r \) is the common ratio, - \( n \) is the term number. In this case, we want to find the third term, so \( n = 3 \). Substitute the values into the formula: \[ a_3 = x \times \left(\frac{x+4}{x}\right)^{(3-1)} \] Simplify the expression to find the third term in terms of \( x \). Simplify the expression by following steps: - step0: Solution: \(x\left(\frac{\left(x+4\right)}{x}\right)^{3-1}\) - step1: Remove the parentheses: \(x\left(\frac{x+4}{x}\right)^{3-1}\) - step2: Subtract the numbers: \(x\left(\frac{x+4}{x}\right)^{2}\) - step3: Rewrite the expression: \(x\times \frac{\left(x+4\right)^{2}}{x^{2}}\) - step4: Reduce the numbers: \(1\times \frac{\left(x+4\right)^{2}}{x}\) - step5: Multiply: \(\frac{x^{2}+8x+16}{x}\) The third term of the geometric sequence in terms of \( x \) is \( \frac{x^2+8x+16}{x} \).