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2023 WAEC GCE

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

2023 WAEC GCE Mathematics questions and Answers

The third and seventh terms of an Arithmetic Progression (AP) are 3+43 and 3+123, 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, T3=3+43, – The seventh term, T7=3+123. We are required to find the fifth term T5 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: Tn=a+(n1)d,
where: – Tn 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 T3: T3=a+2d=3+43,
so we have: a+2d=3+43.(Equation 1)
– For the seventh term T7: T7=a+6d=3+123,
so we have: a+6d=3+123.(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+123)(3+43),
which simplifies to: 4d=83.
Solve for d: d=834=23.

2023 WAEC GCE Mathematics questions and Answers

Step 4: Substitute d into one of the original equations Substitute d=23 into Equation 1: a+2(23)=3+43,
which simplifies to: a+43=3+43.
Solve for a: a=3.

2023 WAEC GCE Mathematics questions and Answers

Step 5: Find the fifth term T5 Now that we know a=3 and d=23, we can find the fifth term using the formula for the n-th term: T5=a+4d=3+4(23)=3+83.
### Final Answer: The fifth term of the arithmetic progression is 3+83.
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