For an $A.P.$,the $p^{th}$ term is $q$ and the $q^{th}$ term is $p$. Find the general term of the $A.P.$
(N/A) Let the first term of the $A.P.$ be $a$ and the common difference be $d$.
The $n^{th}$ term of an $A.P.$ is given by $T_n = a + (n - 1)d$.
Given: $T_p = a + (p - 1)d = q$ --- $(1)$
Given: $T_q = a + (q - 1)d = p$ --- $(2)$
Subtracting equation $(2)$ from $(1)$:
$(a + (p - 1)d) - (a + (q - 1)d) = q - p$
$(p - 1 - q + 1)d = q - p$
$(p - q)d = -(p - q)$
$d = -1$
Substituting $d = -1$ in equation $(1)$:
$a + (p - 1)(-1) = q$
$a - p + 1 = q$
$a = p + q - 1$
The general term $T_n$ is given by:
$T_n = a + (n - 1)d$
$T_n = (p + q - 1) + (n - 1)(-1)$
$T_n = p + q - 1 - n + 1$
$T_n = p + q - n$
The $n^{th}$ term of an $A.P.$ is given by $T_n = a + (n - 1)d$.
Given: $T_p = a + (p - 1)d = q$ --- $(1)$
Given: $T_q = a + (q - 1)d = p$ --- $(2)$
Subtracting equation $(2)$ from $(1)$:
$(a + (p - 1)d) - (a + (q - 1)d) = q - p$
$(p - 1 - q + 1)d = q - p$
$(p - q)d = -(p - q)$
$d = -1$
Substituting $d = -1$ in equation $(1)$:
$a + (p - 1)(-1) = q$
$a - p + 1 = q$
$a = p + q - 1$
The general term $T_n$ is given by:
$T_n = a + (n - 1)d$
$T_n = (p + q - 1) + (n - 1)(-1)$
$T_n = p + q - 1 - n + 1$
$T_n = p + q - n$
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