QUESTION 2 Use Gauss's Method and answer the questions to the series that follow: NO FORMULAE MAY BE USED IN THIS QUESTION! \( 2.11+2+3+4+\ldots .+1000 \) 2.1.1 How many terms are there in the series? 2.1.2 Calculate the sum of the series by using a method similar to that used in Question 1. \[ \left(\begin{array}{l} 1000+999+998+997+\ldots+503+502+501 \\ +72 \end{array}\right. \] \( 2.22+4+6+8+\ldots+72 \) 2.2.1 How many terms are there in the series? 12 2.2.2 Calculate the sum of the series by using a method similar to that used in Question 1.

Let's solve the problems step by step. ### Problem 2.1: Series \( 2.11 + 2 + 3 + 4 + \ldots + 1000 \) #### 2.1.1: How many terms are there in the series? The series starts at \( 2 \) and ends at \( 1000 \). To find the number of terms, we can use the formula for the number of terms in an arithmetic series: \[ n = \text{last term} - \text{first term} + 1 \] Here, the first term \( a = 2 \) and the last term \( l = 1000 \). Calculating the number of terms: \[ n = 1000 - 2 + 1 = 999 \] So, there are **999 terms** in the series. #### 2.1.2: Calculate the sum of the series. To calculate the sum of the series, we can use the formula for the sum of an arithmetic series: \[ S_n = \frac{n}{2} \times (a + l) \] Where: - \( n = 999 \) (number of terms) - \( a = 2 \) (first term) - \( l = 1000 \) (last term) Calculating the sum: \[ S_n = \frac{999}{2} \times (2 + 1000) = \frac{999}{2} \times 1002 \] Now, let's calculate this: \[ S_n = 999 \times 501 = 499499 \] So, the sum of the series is **499499**. --- ### Problem 2.2: Series \( 2 + 4 + 6 + 8 + \ldots + 72 \) #### 2.2.1: How many terms are there in the series? This series is an arithmetic series where the first term \( a = 2 \), the last term \( l = 72 \), and the common difference \( d = 2 \). To find the number of terms, we can use the formula: \[ n = \frac{l - a}{d} + 1 \] Calculating the number of terms: \[ n = \frac{72 - 2}{2} + 1 = \frac{70}{2} + 1 = 35 + 1 = 36 \] So, there are **36 terms** in the series. #### 2.2.2: Calculate the sum of the series. Using the sum formula for an arithmetic series: \[ S_n = \frac{n}{2} \times (a + l) \] Where: - \( n = 36 \) (number of terms) - \( a = 2 \) (first term) - \( l = 72 \) (last term) Calculating the sum: \[ S_n = \frac{36}{2} \times (2 + 72) = 18 \times 74 \] Now, let's calculate this: \[ S_n = 18 \times 74 = 1332 \] So, the sum of the series is **1332**. --- ### Summary of Answers: - **2.1.1**: 999 terms - **2.1.2**: Sum = 499499 - **2.2.1**: 36 terms - **2.2.2**: Sum = 1332