2016 · Q17
Consider the following statements in respect of the expression Sₙ = n(n+1)/2 where n is an integer:
- 1There are exactly two values of n for which Sₙ = 861.
- 2Sₙ = S₋(ₙ₊₁), and hence for any integer m, we have two values of n for which Sₙ = m.
Which of the above statements is/are correct?
Elementary MathematicsNumber System
- A1 only
- B2 only
- CBoth 1 and 2
- DNeither 1 nor 2
Explanation
Sₙ = 861 gives n(n+1)/2 = 861, i.e. n²+n−1722 = 0, with positive root n = 41; since S_{−n−1} = Sₙ, n = −42 also works — exactly two integer values, so statement 1 is true. Statement 2 is false: e.g. Sₙ = 2 has no integer solution. Hence 1 only, option A.