2.2 Given the arithmetic sequence: \( 3 ; b ; 19 ; 27 ; \ldots \) 2.2.1 Calculate the value of \( b \). 2.2.2 Determine the \( n^{\text {th }} \) term of the sequence. 2.2.3 Calculate the value of the thirtieth term \( \left(T_{30}\right) \) (2) 2.2.4 Calculate the sum of the first 30 terms of the sequence.

Let's solve each part of the problem step by step. ### **Given:** An arithmetic sequence: \( 3 ; b ; 19 ; 27 ; \ldots \) In an arithmetic sequence, the difference between consecutive terms is constant. This difference is known as the **common difference (d)**. --- ### **2.2.1 Calculate the value of \( b \).** **Step 1:** Identify the known terms. - First term (\( a_1 \)) = 3 - Second term (\( a_2 \)) = \( b \) - Third term (\( a_3 \)) = 19 **Step 2:** Use the property of arithmetic sequences. - \( a_2 = a_1 + d \) - \( a_3 = a_2 + d \) **Step 3:** Substitute the known values. 1. From \( a_2 = a_1 + d \): \[ b = 3 + d \quad \text{(Equation 1)} \] 2. From \( a_3 = a_2 + d \): \[ 19 = b + d \] Substitute \( b = 3 + d \) from Equation 1: \[ 19 = (3 + d) + d \\ 19 = 3 + 2d \\ 2d = 16 \\ d = 8 \] **Step 4:** Find \( b \) using \( d \). \[ b = 3 + d = 3 + 8 = 11 \] **Answer:** \( b = 11 \) --- ### **2.2.2 Determine the \( n^{\text{th}} \) term of the sequence.** The general formula for the \( n^{\text{th}} \) term (\( a_n \)) of an arithmetic sequence is: \[ a_n = a_1 + (n - 1)d \] **Given:** - \( a_1 = 3 \) - \( d = 8 \) **Substitute the values:** \[ a_n = 3 + (n - 1) \times 8 \\ a_n = 3 + 8n - 8 \\ a_n = 8n - 5 \] **Answer:** The \( n^{\text{th}} \) term is \( a_n = 8n - 5 \) --- ### **2.2.3 Calculate the value of the thirtieth term \( \left(T_{30}\right) \).** Using the \( n^{\text{th}} \) term formula: \[ a_{30} = 8 \times 30 - 5 = 240 - 5 = 235 \] **Answer:** The thirtieth term \( T_{30} = 235 \) --- ### **2.2.4 Calculate the sum of the first 30 terms of the sequence.** The sum of the first \( n \) terms (\( S_n \)) of an arithmetic sequence is given by: \[ S_n = \frac{n}{2} \times (a_1 + a_n) \] **Given:** - \( n = 30 \) - \( a_1 = 3 \) - \( a_{30} = 235 \) **Substitute the values:** \[ S_{30} = \frac{30}{2} \times (3 + 235) \\ S_{30} = 15 \times 238 \\ S_{30} = 3570 \] **Answer:** The sum of the first 30 terms is \( 3,\!570 \). --- ### **Summary of Answers:** 1. **\( b = 11 \)** 2. **\( n^{\text{th}} \) term:** \( a_n = 8n - 5 \) 3. **Thirtieth term (\( T_{30} \)) = 235** 4. **Sum of the first 30 terms = 3,\!570**