# In an Arithmetic sequence, p^(th) term is q and q^(th) term is p.Show that the n^(th) term is p+q-n.?

Nov 29, 2017

#### Explanation:

If first term of an arithmetic sequence is $a$ and common difference is $d$, ${n}^{t h}$ term of arithmetic sequence is $a + \left(n - 1\right) d$

as ${p}^{t h}$ term is $q$ then

$a + \left(p - 1\right) d = q$ ........(1)

and as ${q}^{t h}$ term is $p$ then

$a + \left(q - 1\right) d = p$ ........(2)

subtracting (2) from (1) we get $\left(p - q\right) d = q - p$ i.e. $d = - 1$

and $a = q - \left(p - 1\right) \times \left(- 1\right) = q + p - 1$

and ${n}^{t h}$ term is $a + \left(n - 1\right) \cdot \left(- 1\right)$

= $q + p - 1 - n + 1$

= $p + q - n$