The first torm of a geometric sequence is 3 and the sum of the second and third term is 60 . Detamine the first three terms and the general teim Th

Given that the first term of a geometric sequence is 3 and the sum of the second and third term is 60, we can denote the first term as \(a_1 = 3\) and the common ratio as \(r\). The second term of a geometric sequence is given by \(a_2 = a_1 \cdot r\) and the third term is \(a_3 = a_1 \cdot r^2\). We are given that the sum of the second and third term is 60, so we have: \[a_2 + a_3 = 60\] Substitute the expressions for \(a_2\) and \(a_3\) in terms of \(a_1\) and \(r\): \[a_1 \cdot r + a_1 \cdot r^2 = 60\] Substitute \(a_1 = 3\): \[3 \cdot r + 3 \cdot r^2 = 60\] Simplify the equation: \[3r + 3r^2 = 60\] Divide the entire equation by 3: \[r + r^2 = 20\] This is a quadratic equation in terms of \(r\). We can solve this equation to find the value of \(r\), and then use it to find the first three terms of the geometric sequence. Let's solve the quadratic equation \(r + r^2 = 20\). Solve the quadratic equation by following steps: - step0: Solve by factoring: \(r+r^{2}=20\) - step1: Move the expression to the left side: \(r+r^{2}-20=0\) - step2: Factor the expression: \(\left(r-4\right)\left(r+5\right)=0\) - step3: Separate into possible cases: \(\begin{align}&r-4=0\\&r+5=0\end{align}\) - step4: Solve the equation: \(\begin{align}&r=4\\&r=-5\end{align}\) - step5: Rewrite: \(r_{1}=-5,r_{2}=4\) The solutions to the quadratic equation \(r + r^2 = 20\) are \(r = -5\) and \(r = 4\). Now, we can find the first three terms of the geometric sequence using the common ratio \(r\). For \(r = -5\): \[a_1 = 3\] \[a_2 = a_1 \cdot r = 3 \cdot (-5) = -15\] \[a_3 = a_1 \cdot r^2 = 3 \cdot (-5)^2 = 75\] For \(r = 4\): \[a_1 = 3\] \[a_2 = a_1 \cdot r = 3 \cdot 4 = 12\] \[a_3 = a_1 \cdot r^2 = 3 \cdot 4^2 = 48\] Therefore, the first three terms of the geometric sequence are: - For \(r = -5\): 3, -15, 75 - For \(r = 4\): 3, 12, 48 The general term of a geometric sequence is given by \(a_n = a_1 \cdot r^{n-1}\), where \(a_n\) is the nth term, \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the term number. So, the general term for \(r = -5\) is \(a_n = 3 \cdot (-5)^{n-1}\) and for \(r = 4\) is \(a_n = 3 \cdot 4^{n-1}\).