Question

if pth term of an AP is q and qth term of an AP is p then rth term​

Answer 1

if pth term in an AP is q and qth term of an AP is p.

Then,

rth term is p+q-r

Answer 2

Answer:

rth term of this A.P will be (p + q - r).

Step-by-step explanation:

We have,

An A.P whose pth term is q and qth term is p.

Now,

Let the first term of the A.P be 'a', and its common difference be 'd'.

But, we know that,

The general formula of a term in an A.P is,

a(nth) = a + (n - 1)d

Here we have,

a(pth) = q

Using the above formula,

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

Thus,

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

Similarly,

a(qth) = p

a + (q - 1)d = p ------ 2

Now,

From eq.1 we get,

a = q - (p - 1)d ------ 3

Putting eq.3 in eq.2 we get,

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

Opening the brackets,

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

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

Taking d common,

d[(q - 1) - (p - 1)] = p - q

d[q - 1 - p + 1] = p - q

d(q - p) = (p - q)

Taking (-1) common,

d[(-1)(p - q)] = (p - q)

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

d = 1/(-1)

d = (-1) ----- 4

Putting d = (-1) in eq.3 we get,

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

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

a = q + (p - 1)

a = q + p - 1

a = p + q - 1 ----- 5

Now, we need to find a(rth)

So,

a(rth) = a + (r - 1)d

But from eq.4 and eq.5 we get,

a(rth) = (p + q - 1) + (r - 1)(-1)

a(rth) = p + q - 1 - r + 1

a(rth) = p + q - r + 1 - 1

a(rth) = p + q - r

Hence,

rth term of this A.P will be (p + q - r).

Hope it helped and believing you understood it........All the best.