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If the five numbers a, x, y, z, b form an arithmetic progression and x + y + z = 15, and if the same five numbers form a harmonic progression with 1/x + 1/y + 1/z = 5/3, then what are the values of a and b?

1. 2 and 8
2. 1 and 9
3. 3 and 7
4. None of these

Correct answer: 1 and 9

Solution

The values of a and b are determined by the properties of arithmetic and harmonic progressions. Given that the average of the arithmetic progression is equal to the middle term, and the conditions provided lead to the conclusion that a = 1 and b = 9 satisfy both the sum of the middle terms and the harmonic condition.

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