If in an A.P., Sₙ = qn² and Sₘ = qm², where Sᵣ denotes the sum of first r terms of the A.P., then $$S_q$$​ equals:

Correct option is C

Given:

Sₙ = qn² and Sₘ = qm²
We need $$S_q.$$​​

Concept used:
Sum of n terms of an A.P.:
Sₙ = n/2 [2a + (n − 1)d]

Solution:
Given Sₙ = qn²,
so Sₙ/n = qn => average of n terms = qn

Using Sₙ = $$\frac{n}{2}$$​ [2a + (n − 1)d]

$$\frac{n}{2}$$​ [2a + (n − 1)d] = qn²
=> 2a + (n − 1)d = 2qn ...(1)

Now, if we put n = 1:
2a + 0 = 2q(1)
=> a = q

Substitute in (1):
2q + (n − 1)d = 2qn
=> (n − 1)d = 2q(n − 1)
=> d = 2q

Now find $$S_q:$$​​
​$$S_q = q(q)² = q³$$​​

Correct answer is (c) q³.