Question

Consider the non-decreasing sequence of positive integers 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5.... in which nth positive number appears n times. Find the remainder when the 2000th term is divided by 4.

• A. 1
• B. 2
• C. 3

Answer 1

The correct option is C 3

Let us see the sequence of the numbers:

Number Last term of the number

1 1

2 3

3 6

4 10

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N $\sum n$

We have to find the value of N for the 2000th term. Using iteration we find that if N = 62, the last term that ends with N is $\frac{1}{2}\times 62\times 63$ = 1953.

Therefore, the next 63 terms are 63. So the 2000th term is 63. So the remainder is 3.