If the first term of a harmonic progression is 5 and the common difference of the corresponding arithmetic progression is 2, what is the second term?
Practice Questions
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Q1
If the first term of a harmonic progression is 5 and the common difference of the corresponding arithmetic progression is 2, what is the second term?
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The first term in the arithmetic progression is 1/5, and the common difference is 2. Therefore, the second term in the harmonic progression is 1/(1/5 + 2) = 1/(2.2) = 5/11.
Questions & Step-by-step Solutions
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Q
Q: If the first term of a harmonic progression is 5 and the common difference of the corresponding arithmetic progression is 2, what is the second term?
Solution: The first term in the arithmetic progression is 1/5, and the common difference is 2. Therefore, the second term in the harmonic progression is 1/(1/5 + 2) = 1/(2.2) = 5/11.
Steps: 6
Step 1: Understand that a harmonic progression (HP) is related to an arithmetic progression (AP). The terms of an HP are the reciprocals of the terms of an AP.
Step 2: Identify the first term of the HP, which is given as 5. The first term of the corresponding AP is the reciprocal of this, so it is 1/5.
Step 3: Note that the common difference of the AP is given as 2.
Step 4: Calculate the second term of the AP. The second term is found by adding the common difference to the first term: (1/5) + 2.
Step 5: Convert 2 into a fraction with a denominator of 5 to add: 2 = 10/5. Now add: (1/5) + (10/5) = 11/5.
Step 6: The second term of the HP is the reciprocal of the second term of the AP. So, take the reciprocal of 11/5: 1/(11/5) = 5/11.
