Question

If \[{{\text{m}}^{{\text{th}}}}\] term of HP is n while \[{{\text{n}}^{{\text{th}}}}\] term is m find its \[{{\text{(m + n)}}^{{\text{th}}}}\] term.

Answer 1

Hint – In order to solve this question we need to know that nth term of HP is inverse of nth term of AP. After using this concept as per the conditions given we will get the answer.

Complete step-by-step answer:
As we know nth term of HP is inverse of nth term of AP.
It is given that \[{{\text{m}}^{{\text{th}}}}\] the term of HP is n.
So, ${{\text{T}}_{\text{m}}}{\text{ = }}\dfrac{{\text{1}}}{{{\text{a + (m - 1)d}}}}{\text{ = n}}$ ……(1)
And ${{\text{T}}_{\text{n}}}{\text{ = }}\dfrac{{\text{1}}}{{{\text{a + (n - 1)d}}}}{\text{ = m}}$ ……(2)
Equation (1) can be written as:
${\text{a + (m - 1)d = }}\dfrac{{\text{1}}}{{\text{n}}}$= a + md – d ……(3)
Equation (2) can be written as:
${\text{a + (n - 1)d = }}\dfrac{{\text{1}}}{{\text{m}}}$= a + nd – d ……(4)
On subtracting equation (4) from (3) we get the new equation as:
a – a +md – nd – d + d = $\dfrac{{\text{1}}}{{\text{n}}}{\text{ - }}\dfrac{{\text{1}}}{{\text{m}}}$
(m - n)d = $\dfrac{{\text{1}}}{{\text{n}}}{\text{ - }}\dfrac{{\text{1}}}{{\text{m}}}$

(m - n)d = $\dfrac{{{\text{m - n}}}}{{{\text{mn}}}}$
Then we get, d = $\dfrac{{\text{1}}}{{{\text{mn}}}}$
On putting the value of d in equation (3) we get the new equation as:

$
  {\text{a + (n - 1)}}\dfrac{{\text{1}}}{{{\text{mn}}}}{\text{ = }}\dfrac{{\text{1}}}{{\text{m}}} \\
  {\text{a = }}\dfrac{{\text{1}}}{{\text{m}}}{\text{ - (n - 1)}}\dfrac{{\text{1}}}{{{\text{mn}}}} \\
  {\text{a = }}\dfrac{{{\text{n - n + 1}}}}{{{\text{mn}}}} \\
  {\text{a = }}\dfrac{{\text{1}}}{{{\text{mn}}}} \\
$

Now we have first term and common difference so now we can find the
(m + n)th term of HP.

${{\text{T}}_{{\text{m + n}}}}{\text{ = }}\dfrac{{\text{1}}}{{{\text{a + (m + n - 1)d}}}}$
On putting the value of a and d in above equation we get,
$
  {{\text{T}}_{{\text{m + n}}}}{\text{ = }}\dfrac{{\text{1}}}{{\dfrac{{\text{1}}}{{{\text{mn}}}}{\text{ + (m + n - 1)}}\dfrac{{\text{1}}}{{{\text{mn}}}}}} \\
  {{\text{T}}_{{\text{m + n}}}}{\text{ = }}\dfrac{{\text{1}}}{{\dfrac{{{\text{1 + m + n - 1}}}}{{{\text{mn}}}}}} \\
  {{\text{T}}_{{\text{m + n}}}}{\text{ = }}\dfrac{{{\text{mn}}}}{{{\text{m + n}}}} \\
$
So, we get the \[{\left( {{\text{m + n}}} \right)^{{\text{th}}}}\] term of the HP.

Note – Whenever you face such types of problems you have used the concept that nth term of HP is inverse of nth term of AP. Here in this question we have made equations of AP with the help of given HP then solved it to find the first term and common difference then you can find any of the terms with the help of first term and common difference. Proceeding like this will take you to the right solution of the question asked.