If (S_{n+1}-S_{n-1}=151), what will be the value of (n)?

Correct answer: (75)

Let \\(S_k\\) denote the sum of the first \\(k\\) natural numbers. The difference \\(S_{n+1}-S_{n-1}\\) contains exactly the terms that are present in the first sum but absent from the second. Those two terms are \\(n\\) and \\(n+1\\). Therefore, \\(S_{n+1}-S_{n-1}=n+(n+1)=2n+1\\). The question gives this difference as 151, so \\(2n+1=151\\). Subtracting 1 gives \\(2n=150\\), and dividing by 2 gives \\(n=75\\). Hence option B is correct.

The same result follows from the formula \\(S_k=k(k+1)/2\\), but cancellation of the common terms is simpler and less error-prone. The first sum ends at \\(n+1\\), while the second ends at \\(n-1\\), so only two consecutive terms remain. Values 74 and 76 would make the difference 149 and 153 respectively, not 151. Thus, the supplied answer is mathematically consistent.