(i) The sum of the first three terms of an arithmetic series is 24 and the sum of the next three terms is -12 . Calculate the sum of the first twenty terms of this series. (i) The sum of the first three terms of a geometric series is equal to 15 and the sum of the next three terms is equal to \( -\frac{15}{8} \). Calculate the sum of the first 8 terms of this series. (k) The second term of a convergent geometric series is -12 and the sum to infinity is \( \frac{192}{5} \). Determine the first term and the constant ratio. (1) In a convergent geometric series, the sum of the first four terms is \( \frac{80}{3} \) and the sum to infinity of the series is 27 . Determine the general term.

In an arithmetic series, we first note that the first three terms can be expressed as \(a\), \(a+d\), and \(a+2d\). If their sum is 24, we have \(3a + 3d = 24\), simplifying to \(a + d = 8\). The sum of the next three terms is \((a+3d) + (a+4d) + (a+5d) = 3a + 12d = -12\), which gives us another equation. Solving these two will help us find \(a\) and \(d\), leading to the sum of the first twenty terms using \(S_n = \frac{n}{2}(2a + (n-1)d)\). In the case of the geometric series, the first three terms can be represented as \(a\), \(ar\), and \(ar^2\). The equation for their sum, \(S_3 = a(1 + r + r^2)\), lets us establish a relationship when the sum of those three terms equals 15. For the next three terms, \(ar^3\), \(ar^4\), and \(ar^5\), their sum can also be expressed using the formula for geometric sums. By solving these, you'll uncover \(a\) and \(r\), leading you to find the sum of the first eight terms with \(S_8 = a \frac{1 - r^8}{1 - r}\).