In an A.P., pth term is q and qth term is p, then its common differen

1.	In an A.P., pth term is q and qth term is p, then its common difference is ………………A) 1 B) -1 C) 2 D) -2
Answer» Correct option is (B) -1 Given that \(p^{th}\) term in A.P. is q and \(q^{th}\) term in A.P. is p. i.e., in A.P. \(a_p=q\;\&\;a_q=p\) \(\Rightarrow\) a + (p - 1) d = q _____________(1) and a + (q - 1) d = p _____________(2) \((\because a_n=a+(n-1)d)\) Subtract equation (2) from (1), we obtain (a + (p - 1) d) - (a + (q - 1) d) = q - p \(\Rightarrow(p-1-(q-1))d=q-p\) \(\Rightarrow(p-q-1+1)d=q-p\) \(\Rightarrow(p-q)d=-(p-q)\) \(\Rightarrow d=\frac{-(p-q)}{p-q}=-1\) Hence, common difference of A.P. is -1. Correct option is B) -1

In an A.P., pth term is q and qth term is p, then its common difference is ………………A) 1 B) -1 C) 2 D) -2

Answer»

Correct option is (B) -1

Given that \(p^{th}\) term in A.P. is q and \(q^{th}\) term in A.P. is p.

i.e., in A.P. \(a_p=q\;\&\;a_q=p\)

\(\Rightarrow\) a + (p - 1) d = q _____________(1)

and a + (q - 1) d = p _____________(2) \((\because a_n=a+(n-1)d)\)

Subtract equation (2) from (1), we obtain

(a + (p - 1) d) - (a + (q - 1) d) = q - p

\(\Rightarrow(p-1-(q-1))d=q-p\)

\(\Rightarrow(p-q-1+1)d=q-p\)

\(\Rightarrow(p-q)d=-(p-q)\)

\(\Rightarrow d=\frac{-(p-q)}{p-q}=-1\)

Hence, common difference of A.P. is -1.

Correct option is B) -1