Question

If m time the mth term of an A.P. is equal to n time its nth term, show that the (m + n) term of the A.P. is zero.
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Answer 1

EXPLANATION.

M times the Mth term of an AP is equal to N tines the Nth term of an AP.

Show that the ( m + n ) terms of the AP is = 0.

Formula of Nth term of an AP.

⇒ An = a + ( n - 1 )d.

⇒ M[Tm] = N[Tn].

⇒ m[ a + ( m - 1 )d ] = n[ a + ( n - 1 )d.

⇒ m[ a + md - d] = n[ a + nd - d].

⇒ ma + m²d - md = na + n²d - nd.

⇒ ma + m²d - md - na - n²d + nd = 0.

⇒ ( ma - na ) + ( m²d - n²d ) - ( md - nd ) = 0.

⇒ a ( m - n) + d ( m² - n² ) - d ( m - n ) = 0.

⇒ a ( m - n ) + d ( m + n )( m - n ) - ( m - n ) = 0.

⇒ ( m - n ) [ a + ( m + n - 1 )d ] = 0

⇒ ( m - n ) = 0.

⇒ [ a + ( m + n - 1 ) d ] = 0.

⇒ T(m + n ) = 0.

HENCE PROVED.

MORE INFORMATION.

(1) = Nth term = a + ( n - 1 )d.

(2) = sum of Nth term = Sn = n/2 [ 2a + ( n - 1 )d].

(3) = first term = a.

(4) = common difference = d = b -a = c - b.

(5) = condition of an AP  ⇒ 2b = a + c.