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)

Posted from my mobile device