If p times the pth term of an AP is q times the qth term, then what is the (p + q)th term equal to ?

• A) p + q
• B) pq
• C) 1
• D) 0

In an arithmetic progression (AP), the nth term can be expressed as:

Formula for the nth Term

The nth term of an AP is given by:

Tn = a + (n - 1)d

where a is the first term and d is the common difference.

Setting Up the Equation

According to the problem, we have:

p * Tp = q * Tq

Substituting the formula for the nth term:

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

Expanding the Equation

Expanding both sides gives:

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

Rearranging Terms

Rearranging the equation leads to:

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

This simplifies to:

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

Finding the (p + q)th Term

Now, we need to find the (p + q)th term:

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

Conclusion

From the earlier equation, if we assume that a and d are such that the left side equals zero, we can conclude that:

T(p + q) = 0

Thus, the answer is:

D) 0