Three numbers 5, p and 10 are in Harmonic progression if p = ?

CONCEPT : 

Three numbers x, y, and z are in H.P if and only if y = \(\frac{2xz}{x+z}\)

So, \(\rm {1\over x}\ , {1\over y} \ and \ {1\over z}\) are in A.P. if and only if: \(\frac{2}{y}=\frac{1}{x}+\frac{1}{z}\)

CALCULATION:

Given: Three numbers 5, p and 10 are in Harmonic progression

⇒ \(\rm {1\over 5}\ , {1\over p} \ and \ {1\over 10}\) are in A.P

Now, According to the concept used

⇒ 2 ⋅\(\rm {1\over p}\) = \(\rm {1\over 5} + \rm {1\over 10}\)

⇒ 2 ⋅\(\rm {1\over p}\) = \(\rm \frac {2\ + \ 1}{10}\)

⇒ \(\frac{2}{p}=\frac{3}{10}\)

⇒ p = \(\rm 20\over 3\)

∴ The value of p is 20/3.