If m times the mth term is equal to n times the nth term of an A.P. prove that (m+n)th term of A.P is zero

Let the first term of AP = a common difference = d We have to show that (m+n)th term is zero or a + (m+n-1)d = 0

mth term = a + (m-1)d nth term = a + (n-1) d

Given that m{a +(m-1)d} = n{a + (n -1)d} = am + m²d -md = an + n²d – nd = am – an + m²d – n²d -md + nd = 0 = a(m-n) + (m²-n²)d – (m-n)d = 0 = a(m-n) + {(m-n)(m+n)}d -(m-n)d = 0 = a(m-n) + {(m-n)(m+n) – (m-n)} d = 0 = a(m-n) + (m-n)(m+n -1) d = 0 = (m-n){a + (m+n-1)d} = 0 = a + (m+n -1)d = 0/(m-n) = a + (m+n -1)d = 0

Leave a Comment

Your email address will not be published. Required fields are marked *

BOOK

Free Class