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If m times the m th term of an AP is equal to n times its n th term, show that the (m+n) th term of the AP is 0.

Posted by Robin Andrews Chacko 6 years, 11 months ago

CBSE > Class 10 > Mathematics

2 answers

Sachin Kori 6 years, 11 months ago

Let, "a" be the first term and "d" be the common difference ; Sm =Sn; m/2 { 2a + (m -1)d } = n/2 { 2a + (n-1)d } ; 2am + ( m2 - m )d = 2an + ( n2 - n )d ; 2am - 2an + ( m2 -m )d - ( n2 + n )d = 0 ; 2a (m - n) + ( m2 - m - n2 + n )d = 0 ; 2a (m - n) + ( m2 - n2 - ( m -n )d = 0 ; 2a (m - n) + [( m+n ) (m - n ) - (m - n)]d = 0 ; 2a (m - n) + (m - n) [ m+n - 1]d = 0 ; (m - n) [ 2a + (m+n - 1)d] = 0; since,,, m - n is not equal to zero [therefore m is not equal to n] ; therefore, 2a + ( m+n-1)d = 0 _____ eq.1 ; Sm+n = (m+n)/2 [ 2a + (m+n-1)d] ; Sm+n = m+n/2 × 0 [ from eq. 1] ; Sm+n = 0;; hence proved....

Gaurav Seth 6 years, 11 months ago

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

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