In a given A.P.. If the `p`th term is `q` and `q`th term is `p` then show that `n` th term is `p+q-n`

In a given A.P.. If the `p`th term is `q` and `q`th term is `p` then s

1.	In a given A.P.. If the `p`th term is `q` and `q`th term is `p` then show that `n` th term is `p+q-n`
Answer» Let a be the first term and d be the common difference of the given AP. Then, `T_(p) = a + (p-1)d "and" T_(q) = a + (q-1)d.` Now, `T_(p) = q "and" T_(q) = p` (given). `therefore a + (p-1)d =q " "...(i)` `"and" a+ (q-1)d = p " "...(ii)` On subtracting (i) from (ii), we get `(q-p)d = (p-q) rArr d = -1.` Putting d = -1 in (i), we get a = (p+q-1). Thus, a = (p+q-1) and d = -1. `therefore "nth term" = a + (n-1)d = (p+q-1) + (n-1) xx (-1)` = p+q - n. Hence, nth term = (p+q -n).