Question

if pth term of a AP is 1/q and qth term is 1/p, find the ( pq) th term​

Answer 1

Answer :

(pq)ᵗʰ term is 1

Step-by-step explanation :

Given,

• pth term of AP = 1/q
• qth term of AP = 1/p

To find,

• (pq)th term

Solution,

nth term of AP is given by,

⤳ aₙ = a + (n - 1)d

where

a is the first term

d is the common difference

pth term :

aₚ = a + (p - 1)d

1/q = a + (p - 1)d ---[1]

qth term :

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

1/p = a + (q - 1)d ---[2]

Subtract equation [2] from equation [1],

 1/q - 1/p = a + (p - 1)d - [a + (q - 1)d]

 (p - q)/pq = a + pd - d - [a + qd - d]

 (p - q)/pq = a + pd - d - a - qd + d

 (p - q)/pq = pd - qd

 (p - q)/pq = (p - q)d

       d = 1/pq

➟ common difference = 1/pq

Substitute d = 1/pq in equation [1]

 1/q = a + (p - 1)d

 a = 1/q - (p - 1)(1/pq)

 a = 1/q - 1/q + 1/pq

 a = 1/pq

➟ first term = 1/pq

we have to find (pq)th term

  $\sf a_{pq} =\dfrac{1}{pq} + (pq-1)(\dfrac{1}{pq}) \\\\\\ a_{pq} =\dfrac{1}{pq} + pq(\dfrac{1}{pq}) - \dfrac{1}{pq} \\\\\\ a_{pq}=\dfrac{pq}{pq} \\\\\\ a_{pq}=1$

∴ (pq)ᵗʰ term of the AP is 1