The 1st term of an HP is 1/17 and the product of the 2nd and 4th term equals to the products of 5th and 6th term of the HP. Find the 3rd term of the HP.?

The 1st term of an HP is 1/17 .
Hence, for the corresponding AP, the terms is 17 .
Here, a = 17
nth term of an AP is given by T_n = a + (n -1)d
∴ T₂ = 17 + d, T₄ = 17 + 3d
T₅ = 17 + 4d, T₆ = 17 + 5d
∴ (17 + d) (17 + 3d) = (17 + 4d) (17 + 5d)
⇒ 289 + 51d + 17d + 3d² = 289 + 85d + 68d + 20d²
⇒ 68d + 3d² = 153d + 20d²
⇒ 17d² = -85d
⇒ d = -5
∴ 3rd term is T₃ = a + 2d = 17 + 2(-5)
= 17 - 10 = 7
Hence, for the corresponding HP, the 3rd term is 1/7.