Question

If the 7th term of a H.P. is $\dfrac{1}{{10}}$ and the 12th term is$\dfrac{1}{{25}}$ , then the 20th term is:
A. \[\dfrac{1}{{37}}\]
B. \[\dfrac{1}{{41}}\]
C. \[\dfrac{1}{{45}}\]
D. \[\dfrac{1}{{49}}\]

Answer 1

Hint: We know that HP is the reciprocal of AP. So, to find the 20th term of an HP, we just make use of the reciprocal of the nth term of an AP and find the nth term of the HP.

Complete step by step answer:
It is given that the 7th term of H.P is $\dfrac{1}{{10}}$ therefore,
${T_7} = \dfrac{1}{{10}}$
We can use the following formula to find the value of ‘d’ in this question,
nth term of H.P =$\dfrac{1}{{a + (n - 1)d}}$
$\dfrac{1}{{10}} = \dfrac{1}{{a + \left( {7 - 1} \right)d}} \Rightarrow a + 6d = 10$ …..(1)
Similarly, we have been given with 12th term to be equal to \[\dfrac{1}{{25}}\], therefore,
\[{T_{12}} = \dfrac{1}{{25}}\]
$\dfrac{1}{{25}} = \dfrac{1}{{a + \left( {12 - 1} \right)d}}$
$\dfrac{1}{{25}} = \dfrac{1}{{a + 11d}} \Rightarrow a + 11d = 25$…..(2)
Subtracting (2) from (1), we get,
-5d = -15
    d = 3
Putting d = 3 in equation (1), we get,
$a + 6d = 10$
$a + 6 \times 3 = 10$
$a = 10 - 18$
$a = - 8$
And hence to find the value of 20th term we are going to use the same formula used above that is of nth term, therefore,
${T_{20}} = \dfrac{1}{{ - 8 + \left( {20 - 1} \right) \times 3}}$
\[{T_{20}} = \dfrac{1}{{ - 8 + 19 \times 3}}\]
\[{T_{20}} = \dfrac{1}{{ - 8 + 57}}\]
\[{T_{20}} = \dfrac{1}{{49}}\]

Therefore, option D is the correct answer.

Note: In these questions, start by taking the values given and generating a generalised equation, in this question we had 2 equations and 2 variables, by applying arithmetic operations we could obtain the values of a and d and using that we found the answer.