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If a H.P. whose 5th term is \(\frac{1}{{10}}\) and 10th term is \(\frac{1}{{25}}\). Find the nth term.


1. \(\frac{1}{{ - 2 + 3(n - 1)}}\)
2. \(\frac{1}{{ 2 + 3(n - 1)}}\)
3. \(\frac{1}{{ - 2 + 3(n + 1)}}\)
4. \(\frac{-1}{{ - 2 + 3(n - 1)}}\)

1 Answer

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Best answer
Correct Answer - Option 1 : \(\frac{1}{{ - 2 + 3(n - 1)}}\)

CONCEPT:

A sequence a1, a2, a3,…, an of non-zero numbers is called a Harmonic Progression (HP) if the sequence \(\frac{1}{{{a_1}}},\frac{1}{{{a_2}}},\frac{1}{{{a_3}}}.......,\frac{1}{{{a_n}}}\)is an A.P.

The nth term of the Harmonic Progression (H.P) \( = \frac{1}{{a + (n - 1)d}}\) where 'a' is the first term of A.P d is the common difference and n is the number of terms in A.P.

CALCULATION:

We know that the general form of H.P. is \(\frac{1}{a},\frac{1}{{a + d}},\frac{1}{{a + 2d}}.......\frac{1}{{a + (n - 1)d}}\)

The 5th term is \(\frac{1}{{a + 4d}} = \frac{1}{{10}} \Rightarrow a + 4d = 10\)

The 10th term is \(\frac{1}{{a + 9d}} = \frac{1}{{25}} \Rightarrow a + 9d = 25\)

Solving these two equations, a = -2, d = 3.

Hence the nth term of this HP \( = \frac{1}{{a + (n - 1)d}} = \frac{1}{{ - 2 + 3(n - 1)}}\)

Therefore option (1) is the correct answer.

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