Question

If the m tern of an ap is1/n and the n therm is 1/m show that the sum of mn therm is 1/2 (mn+1)

Answer 1

Hey there!
n-th term of A. P = a + ( n - 1 ) d.

Now,
m term = a + ( m - 1 ) d = 1/n
=> an + ( m - 1 )dn = 1 ...Equation(1)

n term = a + (n-1) d = 1/m

=> am + (n-1)dm = 1 .... Equation(2)

Now, Equation (2) - (1)
am + mnd - md - ( an + mnd - nd) = 1 - 1

am + mnd - md - an - mnd + nd = 0

a ( m - n) - d(m-n) = 0

-d(m-n) = -a(m-n)

$\boxed{a = d}$

From equation 2 ,

am + (m-1)am = 1
am + amn - am = 1
amn = 1
a = 1/mn = d

$\textbf{ Sum of first mn terms} \\ \\= \frac{mn}{2}( 2a + (mn-1)d ) \\ \\ = \frac{mn}{2}(2a+(mn-1)a) \\ \\ = \frac{mn}{2} (2a+amn-a) = \frac{mn}{2}(a + amn) \\ \\ = \frac{mn}{2}(a+ \frac{mn}{mn}) \\ \\ = \frac{mn/2}{a+1} \\ \\ = \frac{mn(a) + mn}{2} \\ \\ = \frac{\frac{mn}{mn}+ mn}{2} \\ \\ = \frac{1+mn}{2}$

Hence proved!