The third and seventh terms of an Arithmetic Progression (AP) are ( 3 + 4\sqrt{3} ) and ( 3 + 12\sqrt{3} ), respectively. Find the fifth term of the A.P.

2023 WAEC GCE Mathematics questions and Answers

The third and seventh terms of an Arithmetic Progression (AP) are \( 3 + 4\sqrt{3} \) and \( 3 + 12\sqrt{3} \), respectively. **Find the fifth term of the A.P.**

2023 WAEC GCE Mathematics questions and Answers

We are given two terms of an arithmetic progression (AP): – The third term, \( T_3 = 3 + 4\sqrt{3} \), – The seventh term, \( T_7 = 3 + 12\sqrt{3} \). We are required to find the fifth term \( T_5 \) of the arithmetic progression.

2023 WAEC GCE Mathematics questions and Answers

Step 1: General formula for the \( n \)-th term of an AP The general formula for the \( n \)-th term of an arithmetic progression is: \[ T_n = a + (n – 1)d, \] where: – \( T_n \) is the \( n \)-th term, – \( a \) is the first term of the AP, – \( d \) is the common difference.

2023 WAEC GCE Mathematics questions and Answers

Step 2: Set up equations for the third and seventh terms Using the general formula for the third and seventh terms: – For the third term \( T_3 \): \[ T_3 = a + 2d = 3 + 4\sqrt{3}, \] so we have: \[ a + 2d = 3 + 4\sqrt{3}. \quad \text{(Equation 1)} \] – For the seventh term \( T_7 \): \[ T_7 = a + 6d = 3 + 12\sqrt{3}, \] so we have: \[ a + 6d = 3 + 12\sqrt{3}. \quad \text{(Equation 2)} \]

2023 WAEC GCE Mathematics questions and Answers

Step 3: Solve the system of equations Subtract Equation 1 from Equation 2 to eliminate \( a \): \[ (a + 6d) – (a + 2d) = (3 + 12\sqrt{3}) – (3 + 4\sqrt{3}), \] which simplifies to: \[ 4d = 8\sqrt{3}. \] Solve for \( d \): \[ d = \frac{8\sqrt{3}}{4} = 2\sqrt{3}. \]

2023 WAEC GCE Mathematics questions and Answers

Step 4: Substitute \( d \) into one of the original equations Substitute \( d = 2\sqrt{3} \) into Equation 1: \[ a + 2(2\sqrt{3}) = 3 + 4\sqrt{3}, \] which simplifies to: \[ a + 4\sqrt{3} = 3 + 4\sqrt{3}. \] Solve for \( a \): \[ a = 3. \]

2023 WAEC GCE Mathematics questions and Answers

Step 5: Find the fifth term \( T_5 \) Now that we know \( a = 3 \) and \( d = 2\sqrt{3} \), we can find the fifth term using the formula for the \( n \)-th term: \[ T_5 = a + 4d = 3 + 4(2\sqrt{3}) = 3 + 8\sqrt{3}. \] ### Final Answer: The fifth term of the arithmetic progression is \( \boxed{3 + 8\sqrt{3}} \).