Q) If p^th term of an A.P. is q and q^th term is p, then prove that its n^th term is (p + q – n).

Ans:

Step 1: We know that n^th term of an A.P. is given by: T_n  =  a + (n -1) d

Therefore, p^th term, T_p = a + (p – 1) x d = q ……….. (i)

Similarly, q^th term, T_q = a + (q – 1) x d = p ………….. (ii)

Step 2: By deducting equation (ii) from equation (i), we get:

[a + (p – 1) d ] – [a + (q – 1) d] = q – p

a + (p – 1) d – a – (q – 1) d = q – p

(p -1) d – (q – 1) d = q – p

pd – d – qd + d = q – p

pd – qd = q – p

d ( p – q) = -1 (p – q)

d = -1 (p – q) / (p – q)

d = -1

Step 3: Let’s find teh value of a from equation (i):

a + (p – 1) d = q

a + (p – 1) ( – 1) = q

a – (p -1) = q

a = q + (p – 1)

a = p + q – 1   

Step 4: Hence, n^th term of this AP, 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 – 1 – n + 1

T_n = p + q – n ……………. Hence Proved

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