If the first term of a harmonic progression is 1 and the second term is 1/3, what is the third term?

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Q: If the first term of a harmonic progression is 1 and the second term is 1/3, what is the third term?

Solution: The reciprocals of the terms are 1, 3, and 1/x. The common difference is 2, so 1/x = 3 + 2 = 5, thus x = 1/5.

Steps: 9

Step 1: Identify the first term of the harmonic progression (HP), which is given as 1.
Step 2: Identify the second term of the HP, which is given as 1/3.
Step 3: Find the reciprocals of the terms. The first term's reciprocal is 1/1 = 1, and the second term's reciprocal is 1/(1/3) = 3.
Step 4: Let the third term of the HP be x. The reciprocal of the third term will be 1/x.
Step 5: Since the terms are in harmonic progression, the reciprocals (1, 3, and 1/x) form an arithmetic progression (AP).
Step 6: In an AP, the difference between consecutive terms is constant. Calculate the common difference: 3 - 1 = 2.
Step 7: Set up the equation for the third term's reciprocal: 1/x = 3 + 2.
Step 8: Simplify the equation: 1/x = 5.
Step 9: Solve for x by taking the reciprocal: x = 1/5.