If the first term of a harmonic progression is 5 and the second term is 10, what

If the first term of a harmonic progression is 5 and the second term is 10, what is the fourth term?

If the first term of a harmonic progression is 5 and the second term is 10, what is the fourth term?

Step 1: Identify the first term of the harmonic progression (HP), which is given as 5.
Step 2: Identify the second term of the harmonic progression (HP), which is given as 10.
Step 3: Find the reciprocals of the first and second terms. The reciprocal of 5 is 1/5, and the reciprocal of 10 is 1/10.
Step 4: Calculate the common difference between the reciprocals. Subtract 1/5 from 1/10: 1/10 - 1/5.
Step 5: To subtract, convert 1/5 to a fraction with a denominator of 10. 1/5 = 2/10.
Step 6: Now subtract: 1/10 - 2/10 = -1/10. This is the common difference.
Step 7: To find the fourth term, we need to find the reciprocal of the third term first. The third term's reciprocal is 1/10 - 1/10 = 0, which is not valid, so we need to continue with the common difference.
Step 8: The third term's reciprocal is 1/10 - 1/10 = 0, which means we need to find the fourth term's reciprocal by continuing the pattern.
Step 9: The fourth term's reciprocal is 1/10 - 2 * (1/10) = 1/10 - 2/10 = -1/10, which is incorrect. Let's correct this.
Step 10: The correct approach is to find the fourth term's reciprocal by adding the common difference to the second term's reciprocal: 1/10 - 1/10 = 1/25.
Step 11: The fourth term is the reciprocal of 1/25, which is 25.

Harmonic Progression – A sequence of numbers is in harmonic progression if the reciprocals of the terms form an arithmetic progression.
Reciprocal Relationships – Understanding how to manipulate and calculate the reciprocals of terms in a harmonic progression.
Common Difference – The difference between consecutive terms in the arithmetic progression formed by the reciprocals.