Question

If the pth term of a arithmetic series is 1 / q and qth term is 1 / p, prove that pqth term will be 1.​

Answer 1

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Answer 2

Answer:

Step-by-step explanation:

nth term of AP is given by tₙ = a + (n - 1)d

pth term of AP, tp = a + (p - 1)d = 1/q --- (1)

qth term of AP, tq = a + (q - 1)d = 1/p --- (2)

Now (1) - (2),

1/q - 1/p = (p - q)d

=> p - q/pq = (p - q)d

=> d = 1/pq --- (3)

Substitute value of d in (1)

a + (p - 1)1/pq = 1/q

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

pqth term of AP is given by tpq = a + (pq - 1)d

                                                     = 1/pq + (pq - 1)1/pq

                                                     = 1/pq + 1 - 1/pq = 1

Hence proved.