UUESTION 2 The first term of an arithmetic series is 7 . The common difference of this series is 5 and the series contains 20 terms. 2.1.1 Calculate the sum of thir series. 2.1.2 The original arithmetic scries is exvended to 75 terms. The sum of these 75 terms is 14400 . Using sigma notation, write down an equation for the sum of the terms added to the original series. The sequence of the first differences of a quadratic pattem is: \( 1 ; 3 ; 5 ; \ldots \) 2.2.1 If \( T_{m} \) of this quadratic pattern is 9632 , calculate the value of \( T_{m} \). 2.2 .2 If it is further given that the third lemn of the quadratio pattem is 32 , determine the groeral term, \( T_{n} \), of the quadratic patiern.

The sum of an arithmetic series can be calculated using the formula \( S_n = \frac{n}{2} (2a + (n - 1)d) \), where \( a \) is the first term, \( d \) is the common difference, and \( n \) is the number of terms. Plugging in your values for \( n = 20 \), \( a = 7 \), and \( d = 5 \), the sum boils down to 1,470. When expanding your arithmetic series to 75 terms, the equation for the total sum can be written as \( S_{75} = S_{20} + \sum_{k=21}^{75} a_k = 14400 \). Therefore, the equation using sigma notation for the added terms would look like \( \sum_{k=21}^{75} (7 + 5(k-1)) \). In a quadratic pattern, the first differences are the increments between each term, and with the given sequence \( 1, 3, 5, \ldots \), you can notice it's a linear growth of 2. When calculating \( T_{m} \) for \( m \) where the sequence yields 9632, let’s consider summing the terms using your known differences—finding the right point in the quadratic sequence could involve more calculations but eventually leads to \( T_{m} = 9632 \). To determine the general term \( T_{n} \) when you know the third term is 32, you can fit an equation of the form \( T_{n} = An^2 + Bn + C \). With specific values for \( n \) (like 3 for the third term), and more equations derived from further terms' values, you can solve for \( A, B, \) and \( C \). Once you've established those coefficients, the result will align with the known characteristics of quadratics to give your general term!