Question

If mth term of AP is n and nth term of AP is m, show that (m+n)th term of the AP vanishes.​

Answer 1

Correct Question :

If mth term of AP is n and nth term of AP is m, what is the (m + n)th term of the AP?

Solution :

Given that,

• $\sf a_m = n$
• $\sf a_n = m$

Nth term of an AP is described as :

$\sf T_N = a + (N - 1)d$

Likewise,

$\sf a_m = a + (m - 1)d \\ \\ \dashrightarrow \sf \: a + (m - 1)d = n - - - - - - (1)$

Similarly,

$\sf a_n = a + (n - 1)d \\ \\ \dashrightarrow \sf \: a + (n - 1)d = m - - - - - - (2)$

Solving equations (1) and (2), we get common difference and first term of the above AP.

Subtracting (2) and (1), we get :

$\sf \: md - d - nd + d = n - m \\ \\ \longrightarrow \sf \: (m - n)d = (n - m) \\ \\ \longrightarrow \boxed{ \boxed{ \sf \: d = - 1 }}$

Substitute d = - 1 in any of the equations to get value of first term (let equation 1),

$\sf \: a + (m - 1)( - 1) = n \\ \\ \longrightarrow \sf \: a - m + 1 = n \\ \\ \longrightarrow \boxed{\boxed{ \sf \: a = m + n - 1 }}$

Now, (m + n)th term will be :

$\sf \: a_{m + n} = m + n - 1 + (m + n - 1)( - 1) \\ \\ \longrightarrow \sf \: a_{m + n} = m + n - 1 - m - n + 1 \\ \\ \longrightarrow \boxed{ \boxed{ \sf \: a_{m + n} =0}}$

Answer 2

Answer:

m+n th term of  ap = 0

Step-by-step explanation:

if mth term of an A.P.is n and nth term is m, show that (m+n)th term vanishes

mth term of an A.P.  = a + (m-1)d = n  => a = n -(m-1)d  

nth Term of an ap =  a + (n-1)d = m  => a = m - (n-1)d  

n -(m-1)d = m - (n-1)d  

=> n - m = d(m-1 -n + 1)  

=> n - m = d(m-n)  

=> d = -1  

Puttind d = -1  

a = n + m - 1    

m+n th term of  ap = a + (m + n - 1)d

putting a = m+n-1  & d = -1

= n + m - 1 + (m + n  -1)(-1)

= n + m - 1 -m -n + 1  

= 0

m+n th term of  ap = 0

thanks.