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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.
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2023 WAEC GCE Mathematics questions and Answers
The third and seventh terms of an Arithmetic Progression (AP) are 3+4√3 and 3+12√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, T3=3+4√3, – The seventh term, T7=3+12√3. 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+(n–1)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+4√3,
so we have:
a+2d=3+4√3.(Equation 1)
– For the seventh term T7:
T7=a+6d=3+12√3,
so we have:
a+6d=3+12√3.(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√3)–(3+4√3),
which simplifies to:
4d=8√3.
Solve for d:
d=8√34=2√3.
2023 WAEC GCE Mathematics questions and Answers
Step 4: Substitute d into one of the original equations Substitute d=2√3 into Equation 1: a+2(2√3)=3+4√3,
which simplifies to:
a+4√3=3+4√3.
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=2√3, we can find the fifth term using the formula for the n-th term: T5=a+4d=3+4(2√3)=3+8√3.
### Final Answer:
The fifth term of the arithmetic progression is 3+8√3.
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