Given 𝑆𝑁 represents the sum of 𝑁 terms of a certain sequence. It is said that the sequence is obtained by adding a positive constant C to the previous term, which means it is in Arithmetic Progression.
As we know, Arithmetic Progression is a sequence obtained by adding or subtracting a constant from the previous terms. This leads to a common difference between each of the two consecutive terms.
If 𝐴 be the first term and 𝐷 be the common difference then the sequence proceed as follows:
1st, 2nd, 3rd, …………………………………., 𝑁th term
𝐴, 𝐴 + 𝐷, 𝐴 + 2𝐷, 𝐴 + 3𝐷, … … … , 𝐴 + (𝑁 − 1)𝐷
Where 𝑁 is the last term.
Let 𝐴 be the first term. The common difference here is 𝐷.
So, 𝑆𝑛 can be expressed as:
𝑆𝑁 = [𝐴] + [𝐴 + 𝐷] + [𝐴 + 2𝐷] + ⋯ + [𝐴 + (𝑁 − 1)𝐷] = Sum of 𝑁 terms
Accordingly, 𝑆𝑁+1 can be expressed as:
𝑆𝑁+1 = 𝑆𝑢𝑚 𝑜𝑓 (𝑁 + 1) 𝑇𝑒𝑟𝑚s
Similarly, 𝑆𝑁+2 can be expressed as:
𝑆𝑁+2 = 𝑆𝑁+1 + extra term
So in both quantities everything boils down to the extra term, considering also that Sn+1 is common to both and this means we have 0
QA is 0
QB the extra term.
However, we do not know the extra term if it is positive, negative, or zero
D is the answer
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