JEE MainMathematicsEasySequences and Series

Find the number of positive integral ordered pairs of (x, y) so
that 8, x, y are in harmonic progression.

Show answer and solution

8, x, y are in H.P
Hence

x = \frac{16y}{8 + y},x = 16\left( 1 - \frac{8}{y + 8} \right)

x = 16 - \frac{128}{y + 8}

Where

\frac{128}{y + 8} < 16

\Rightarrow

y+8 > 8 and 128 is divisible by (y + 8)

\therefore

y + 8 = 16, 32, 64, 128
4 values of y
So, 4 ordered pairs

Source: Grass-G / M-23 Mathematics Paper 31 Dec FST.docx

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