if m times mth term of an Ap is equal to n times nth term, find its (m+n)th term...

PLZ ANSWER IT SOON......

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

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

equation (i)-(ii)

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

d = -1 _______________(iii)

putting vaiue of d in eqⁿ (i)

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

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

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

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

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

a_(m+n) = 0

Let a be the first terms and d be the common difference of given A.P.

Then, a_m = a + (m – 1) d and

a_n = a + (n – 1) d

According to given equation,

m × a_m = n × a_n

⇒ m × [a + (m – 1) d] = n [a + (n – 1) d]

⇒ am + dm (m – 1) = an + dn (n – 1)

⇒ am – an = dn (n – 1) – dm (m – 1)

⇒ a(m – n) = d(n ² – n – m ² + m)

Now, a_{m + n} = a + (m + n – 1) d

⇒ (1 – m – n)d + (m + n – 1)d [Using (1)]

⇒ d – md – nd + md + nd – d

⇒ 0

Hence, the (m + n)th terms of given AP is zero

ANSWER OF EXPERT

am = n

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

an = m

a + ( n - 1 ) d = m ... ( 2 )

(1) - (2)

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

a - a + md - d - { nd - d } = n - m

a - a + md - d - nd + d = n - m

d ( m- n ) = n - m

n - m = -1 ( m - n )

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

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

d = -1

a + ( m - 1 ) d = n

a + ( m - 1 ) -1 = n

a + -m +1 = n

a = n + m - 1

to prove: (m+n)th term is zero.

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

here , n = no. of terms

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

= n + m - 1 - m - n + 1

= 0

same as above