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

But it is mentioned that c>0; will that not make S(n+2) bigger?

C> 0 is what in the end create the sequence

Turns out the sequence is a term + another one

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.

So you have Sn+1+another term = Sn+2

In other words, we do have Sn+1+extra term=Sn+2+extra term=Sn+3 and so forth

Therefore in both quantities, Sn+1 is equal

Sn+1=Sn+2 because Sn+1=Sn+1 + extra term

Simplify we are left with just the extra term that we do not know is is positive, negative or zero

C>0 is used to create the sequence. It is another thing

Let me know if now is more clear

Carcass
what does c>0 represent here? if it does not mention c should be a +ve value

As I said above C is the term to create the sequence.

See my explanation above.

Based on C the sequence could be different

The way I solved it was to take an example....(the fact that question says c>0, it wouldn't make any sense to assume +ve or -ve values for C..its absurd)

Assume c = 1 (since +ve)
For a series that would end in a negative number,
Eg .... -10, -9, -8, -7, -6 here if nth term is -8 then (n+2)th terms would be adding more -ve numbers to the sum and thus it would be lesser than sum of (n+1)th term....

This would reverse for a positive sequence of numbers.

Hence answer is D ..( rln cannot be determined)

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Okay, but how do you know if the question (quantity A and quantity B) is asking for Sn+1 (the sequence) or Sn+1 (the term)? I don't understand ... The question isn't worded properly imo, if they are asking to compare terms, then the answer would be different

Look at the previous marvelous explanations. Thy are there for a specific reason

Moreover

where each term after the first term of the sequence is obtained by adding a constant

So is the term