Question

If $ mth $ term of an AP is $ \dfrac{1}{n} $ and then $ nth $ term is $ \dfrac{1}{m} $ then find the sum of its first $ mn $ terms.

Answer 1

Hint: As we know that an arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant. Here in this question we have to find the sum of the arithmetic sequence , we know the formula of the sum of arithmetic sequence is $ {S_n} = \dfrac{n}{2}\{ 2a + (n - 1)d\} $ where $ n $ is the number of terms, $ a = $ first term and $ d $ is the common difference. We will first find the common difference between the two terms of the given series and then we will substitute the values in the formula.

Complete step by step solution:
Here we have to find the sum of $ mn $ terms of an AP. Let us assume $ a $ be the first term and $ d $ be the common difference. Now we will use the formula for $ mth $ term. So we have: $ {a_m} = a + (m - 1)d = \dfrac{1}{n} $
Similarly for $ nth $ term which is $ \dfrac{1}{m} $ , we have:
$\Rightarrow {a_n} = a + (n - 1)d = \dfrac{1}{m} $ .
Now we will subtract both the equations and we have,
$\Rightarrow (m - 1)d - (n - 1)d = \dfrac{1}{n} - \dfrac{1}{m} $
Now solving this, $ dm - d - dn + d = \dfrac{{m - n}}{{mn}} \Rightarrow dm - dn = \dfrac{{m - n}}{{mn}} $ .
WE can now take the $ d $ common and then solve:
 $ d(m - n) = \dfrac{{m - n}}{{mn}}\\
 \Rightarrow d = \dfrac{1}{{mn}} $ .
Now we will put the value of $ d $ in the $ {a_m} $ equation, we have;
 $ a + (m - 1) \times \dfrac{1}{{mn}} = \dfrac{1}{n} \Rightarrow a = \dfrac{1}{n} - \dfrac{{m - 1}}{{mn}} $ .
This gives us $ a = \dfrac{1}{{mn}} $ .
So the sum of the first $ mn $ terms be :
 $ {S_{mn}} = \dfrac{{mn}}{2}\left\{ {2\left( {\dfrac{1}{{mn}}} \right) + (mn - 1)\left( {\dfrac{1}{{mn}}} \right)} \right\} $ $ = \dfrac{{mn}}{2}\left[ {\dfrac{1}{{mn}} + 1} \right] $
We can write it as
 $ \left( {\dfrac{{1 + mn}}{{mn}}} \right) \times \dfrac{{mn}}{2} $ ,
it gives us the value $ \left( {\dfrac{{1 + mn}}{2}} \right) $ .
Hence the required answer is $ \left( {\dfrac{{1 + mn}}{2}} \right) $ .
So, the correct answer is “ $ \left( {\dfrac{{1 + mn}}{2}} \right) $ ”.

Note: We should be aware of the arithmetic sequence and their formula before solving this kind of question. We should carefully substitute the values and solve them. Also we can find the sum of the given arithmetic sequence by adding the first and last term and then divide the sum by two, this is also a formula when the first and the last term is given in the question. And then the sum of the sequence will be the number of terms multiplied by the average number of terms in the sequence.