(1) The fifth term of an arithmetic sequence is 0 and thirteenth term is 12 . Determine the sum of the first 21 terms of the sequence. 2) The \( 1^{\text {st }} \) term of an arithmetic sequence is 6 and the sum

To solve this problem, we will first determine the common difference of the arithmetic sequence using the given information. Then, we will find the sum of the first 21 terms of the sequence. Given: - The fifth term of the arithmetic sequence is 0. - The thirteenth term of the arithmetic sequence is 12. Let's denote the first term of the arithmetic sequence as \(a\) and the common difference as \(d\). 1. Determine the common difference: We know that the nth term of an arithmetic sequence is given by the formula: \[a_n = a + (n-1)d\] Given that the fifth term is 0, we have: \[a_5 = a + 4d = 0\] Given that the thirteenth term is 12, we have: \[a_{13} = a + 12d = 12\] Now, we can solve these two equations to find the values of \(a\) and \(d\). 2. Find the sum of the first 21 terms: Once we have the values of \(a\) and \(d\), we can use the formula for the sum of the first n terms of an arithmetic sequence: \[S_n = \frac{n}{2} \times (a + a_n)\] Let's proceed with solving the equations to find the common difference and then calculate the sum of the first 21 terms. Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}a+4d=0\\a+12d=12\end{array}\right.\) - step1: Solve the equation: \(\left\{ \begin{array}{l}a=-4d\\a+12d=12\end{array}\right.\) - step2: Substitute the value of \(a:\) \(-4d+12d=12\) - step3: Simplify: \(8d=12\) - step4: Divide both sides: \(\frac{8d}{8}=\frac{12}{8}\) - step5: Divide the numbers: \(d=\frac{3}{2}\) - step6: Substitute the value of \(d:\) \(a=-4\times \frac{3}{2}\) - step7: Simplify: \(a=-6\) - step8: Calculate: \(\left\{ \begin{array}{l}a=-6\\d=\frac{3}{2}\end{array}\right.\) - step9: Check the solution: \(\left\{ \begin{array}{l}a=-6\\d=\frac{3}{2}\end{array}\right.\) - step10: Rewrite: \(\left(a,d\right) = \left(-6,\frac{3}{2}\right)\) The solution to the system of equations is: \[a = -6\] \[d = \frac{3}{2}\] Now, let's find the sum of the first 21 terms of the arithmetic sequence using the formula: \[S_n = \frac{n}{2} \times (a + a_n)\] Substitute the values of \(a\), \(d\), and \(n\) into the formula to find the sum of the first 21 terms. Calculate the value by following steps: - step0: Calculate: \(\frac{21}{2}\left(-6+\left(-6+\frac{20\times 3}{2}\right)\right)\) - step1: Reduce the fraction: \(\frac{21}{2}\left(-6+\left(-6+30\right)\right)\) - step2: Add the numbers: \(\frac{21}{2}\left(-6+24\right)\) - step3: Add the numbers: \(\frac{21}{2}\times 18\) - step4: Reduce the numbers: \(21\times 9\) - step5: Multiply the numbers: \(189\) The sum of the first 21 terms of the arithmetic sequence is 189.