Question

If mth and nth term of an A.P. are 1/n and 1/m respectively, then find its
mnth
term.​

Answer 1

$\large\underline{\sf{Solution-}}$

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

↝ nᵗʰ term of an arithmetic sequence is,

$\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{a_n\:=\:a\:+\:(n\:-\:1)\:d}}}}}} \\ \end{gathered}$

Wʜᴇʀᴇ,

• aₙ is the nᵗʰ term.

• a is the first term of the sequence.

• n is the no. of terms.

• d is the common difference.

Tʜᴜs,

According to statement,

↝ mᵗʰ term is 1/n.

$\rm :\implies\:a + (m - 1)d = \dfrac{1}{n} - - - (1)$

Also, given that

↝ nᵗʰ term is 1/m.

$\rm :\implies\:a + (n - 1)d = \dfrac{1}{m} - - - (2)$

On Subtracting equation (2) from equation (1), we get

$\rm :\implies\: \cancel{a} + (m - 1)d - \cancel{a} - (n - 1)d = \dfrac{1}{n} - \dfrac{1}{m}$

$\rm :\longmapsto\:d(m - \cancel{1} - n + \cancel{1}) = \dfrac{m - n}{mn}$

$\rm :\longmapsto\:d\cancel{(m - n)} = \dfrac{\cancel{m - n}}{mn}$

$\rm :\implies\:d = \dfrac{1}{mn} - - - (3)$

On substituting the value of d in equation (1), we get

$\rm :\longmapsto\:a + (m - 1)\dfrac{1}{mn} = \dfrac{1}{n}$

$\rm :\longmapsto\:a +\dfrac{\cancel{m}}{\cancel{m} \: n} - \dfrac{1}{mn} = \dfrac{1}{n}$

$\rm :\longmapsto\:a +\cancel{\dfrac{1}{n}} - \dfrac{1}{mn} = \cancel{\dfrac{1}{n}}$

$\rm :\implies\:a = \dfrac{1}{mn} - - - (4)$

Now,

$\rm :\longmapsto\:a_{mn} = a + (mn - 1)d$

On substituting the values of a and d, we get

$\rm :\longmapsto\:a_{mn} = \dfrac{1}{mn} + (mn - 1)\dfrac{1}{mn}$

$\rm :\longmapsto\:a_{mn} = \cancel{\dfrac{1}{mn}} + \dfrac{\cancel{mn}}{\cancel{mn}} - \cancel{\dfrac{1}{mn}}$

$\rm :\longmapsto\:a_{mn} = 1$

$\boxed{ \bf \: Hence, \: a_{mn} = 1}$

Additional Information :-

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

↝ Sum of first n terms of an arithmetic sequence is,

$\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{S_n\:=\dfrac{n}{2} \bigg( 2\:a\:+\:(n\:-\:1)\:d \bigg)}}}}}} \\ \end{gathered}$

Wʜᴇʀᴇ,

• Sₙ is the sum of first n terms.

• a is the first term of the sequence.

• n is the no. of terms.

• d is the common difference.

Answer 2

Bina jaane usse pyaar krti ho

Wah!

One sided love hai syd tumhara,