(b) For a certain series, it is given that \( \sum_{i=1} T_{i}=5 n^{2}-2 n \). (1) Calculate the sum of the first 50 terms of the series. (2) Determine \( T_{20} \). (3) How many terms of the series will add up to 6055 ? (c) The sum to \( n \) terms of a series is given by \( S_{n}=\frac{n}{2}[8 n-7] \). (1) How many terms must be added to get a sum of 9825 ?

To solve the given problems step-by-step: **(b)** Given \( \sum_{i=1}^n T_i = 5n^2 - 2n \): 1. To find the sum of the first 50 terms: \[ S_{50} = 5(50)^2 - 2(50) = 5 \cdot 2500 - 100 = 12500 - 100 = 12400. \] 2. To find \( T_{20} \), we need \( T_{20} = S_{20} - S_{19} \): \[ S_{20} = 5(20)^2 - 2(20) = 5 \cdot 400 - 40 = 2000 - 40 = 1960, \] \[ S_{19} = 5(19)^2 - 2(19) = 5 \cdot 361 - 38 = 1805 - 38 = 1767, \] \[ T_{20} = S_{20} - S_{19} = 1960 - 1767 = 193. \] 3. To find how many terms add up to 6055, set \( S_n = 6055 \): \[ \sum_{i=1}^n T_i = 5n^2 - 2n = 6055. \] Rearranging, we get: \[ 5n^2 - 2n - 6055 = 0. \] Using the quadratic formula: \[ n = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 5 \cdot (-6055)}}{2 \cdot 5} = \frac{2 \pm \sqrt{4 + 121100}}{10} = \frac{2 \pm 348}{10}. \] This gives solutions: \[ n = \frac{350}{10} = 35 \quad \text{(positive root)}. \] Hence, 35 terms of the series will add up to 6055. **(c)** Given the sum to \( n \) terms \( S_n = \frac{n}{2}[8n - 7] \): 1. To find how many terms must be added to get a sum of 9825: \[ \frac{n}{2}(8n - 7) = 9825 \implies n(8n - 7) = 19650. \] Rearranging gives: \[ 8n^2 - 7n - 19650 = 0. \] Using the quadratic formula: \[ n = \frac{-(-7) \pm \sqrt{(-7)^2 - 4 \cdot 8 \cdot (-19650)}}{2 \cdot 8} = \frac{7 \pm \sqrt{49 + 628800}}{16} = \frac{7 \pm \sqrt{628849}}{16}. \] The square root simplifies to 791: \[ n = \frac{7 \pm 791}{16}. \] The positive solution is: \[ n = \frac{798}{16} = 49.875. \] Since \( n \) must be an integer, rounding gives \( n = 50 \). Thus, 50 terms must be added to reach a sum as close as possible to 9825. And there you have it, a thrilling journey through sums and series with numbers dancing across the page! Whether you’re solving for specific terms or total sums, you are wielding the power of mathematics like a pro!