In: Math
If the pth term of an AP is q and the qth term is p, prove that its nth term is (p + q - n).
• pth term = q
• q th term = p
a +(p – 1) d = q______(1)
a +(q – 1) d = p______(2)
Subtracting equation (2) – (1), we get
a +(q – 1) d – [a +(p – 1) d] = p – q
a + dq – d – a – pd – + d = p – q
dq – dp = p – q
d(q – p) = -(q – p)
d = -1
Putting d= -1 in equation (1)
a +(p – 1) d = q
a + (p – 1)(-1) = q
a – p + 1 = q
a = p + q – 1
Thus,
nth term = a + (n – 1)d
nth term = (p + q – 1) + (n – 1) × (-1)
nth term = (p + q – n)
nth term = (p + q – n)