Question

if Pth term is p and Qth term is q then sum of sequenceof nth term is?

Answer 1

let a be the first term and d be the common difference of given AP then
     pth term = q
     a+(p-1)d=q_______________1
     
     qth term=p
     a+(q-1)d=p__________________2

subtracting equ 2 from 1
    (p-q)d=(q-p)
            d= -1
putting d = -1 in equ 1 , we get
       a+(p-1)-1 =q
                  a   = (p+q-1)

 nth term = a+(n-1)d
                 =(p+q-1)+(n-1)×(-1)
                 =(p+q-n)

Answer 2

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$let \: a \: be \: the \: first \: term \: and \: d \: be \: the \: common \: difference \: of \: the \: nth \: term \: of \: ap \\ t _{p} = a + (p - 1)d \: \: and \: t _{q} = a + (q - 1)d \\ now \: t _{p } = q \: and \: t _{q} = p \\ \therefore \: a + (p - 1)d = q \: \: \: \: \: \: \: \: \: \: \: \: ... .(1) \\ and \: a + (q - 1)d = p \: \: \: \: \: \: \: \: \: \: \: \: \: .. .. (2) \\ \\ \\ on \: subtracting \: (1)from(2) \: we \: get \\ (q - p)d = (p - q) \implies \: d = - 1 \\ putting \: d = - 1 \: in \: (1) \: we \: get \: a = (p + q - 1) \\ \therefore \: nth \: term \: = a(n - 1)d = (p + q - 1) + (n - 1)( - 1) = (p + q - n) \\ \\ hence \: nth \: term \: = (p + q - n)$

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