Question

if am=n and an=m then prove am+n =0

Answer 1

$<b>$Correction in question :-

==> am + an =0

Or

a(m+n)=0

${\huge{\mathfrak{Answer:-}}}$

$<u>$

an= a+ (n-1)d ,i.e, m = a+(n - 1)d ----- Eq. 1

am = a+(m-1)d ,i.e, n = a+(m - 1)d ---- Eq. 2

Eq.1 - Eq 2

m-n = d(n-1) - d(m-1)

m-n = d [n-1-(m-1)]

m-n= d[n-1-m+1]

m-n= -d[m-n]

-d =1 i.e d=-1

Substituting in d Eq. 1 :

m = a + (n-1) (-1)

m = a + 1- n , so, a = m+n-1 {eq 3}

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

a(m+n) = (m + n - 1) + (m + n - 1)(-1)

a(m+n) = (m+n-1)(1-1)

a(m+n)= (m+n-1) 0

a(m+n)=0