ERIES - ARITHMETIC SEQ EXAMPLE 2 \( m ; 2 m+2 \) and \( 5 m+3 \) are three consecutive terms of an arithmetic sequence. a) Determine the value of \( m \). b) Determine the sequence.

Let's solve the given problem step by step. **Given:** Three consecutive terms of an arithmetic sequence are: - First term: \( m \) - Second term: \( 2m + 2 \) - Third term: \( 5m + 3 \) An **arithmetic sequence** is a sequence of numbers where the difference between consecutive terms is constant. This constant difference is known as the **common difference** (\( d \)). ### a) Determine the value of \( m \) To find the value of \( m \), we'll set up an equation based on the property of arithmetic sequences. 1. **Find the common difference between the first and second terms:** \[ d = (2m + 2) - m = m + 2 \] 2. **Find the common difference between the second and third terms:** \[ d = (5m + 3) - (2m + 2) = 3m + 1 \] 3. **Since the common difference is the same for both pairs of consecutive terms, set them equal:** \[ m + 2 = 3m + 1 \] 4. **Solve for \( m \):** \[ m + 2 = 3m + 1 \\ 2 - 1 = 3m - m \\ 1 = 2m \\ m = \frac{1}{2} \] **Answer to part (a):** \[ m = \frac{1}{2} \] ### b) Determine the sequence Now that we have the value of \( m \), we can find the three consecutive terms of the arithmetic sequence. 1. **First term (\( a_1 \)):** \[ a_1 = m = \frac{1}{2} \] 2. **Second term (\( a_2 \)):** \[ a_2 = 2m + 2 = 2 \left( \frac{1}{2} \right) + 2 = 1 + 2 = 3 \] 3. **Third term (\( a_3 \)):** \[ a_3 = 5m + 3 = 5 \left( \frac{1}{2} \right) + 3 = \frac{5}{2} + 3 = \frac{11}{2} = 5.5 \] 4. **Common difference (\( d \)):** \[ d = m + 2 = \frac{1}{2} + 2 = \frac{5}{2} = 2.5 \] 5. **General form of the sequence:** \[ a_n = a_1 + (n-1)d \] Plugging in the known values: \[ a_n = \frac{1}{2} + (n-1) \times \frac{5}{2} \] **Answer to part (b):** The three consecutive terms of the arithmetic sequence are: \[ \frac{1}{2}, \quad 3, \quad \frac{11}{2} \] or equivalently: \[ 0.5,\quad 3,\quad 5.5 \] The common difference is \( \frac{5}{2} \) (or 2.5), and the general formula for the \( n \)-th term of the sequence is: \[ a_n = \frac{1}{2} + (n-1) \times \frac{5}{2} \]