Sum of Consecutive Numbers

Q. When the natural numbers 1, 2, 3, ..., 500 are written, then the digit 3 is used n times in this way. The value of n is:

1. 100
2. 200
3. 300
4. 280

Q. A triangular number is defined as a number which has the property of being expressed as a sum of consecutive natural numbers starting with 1. How many triangular numbers less than 1000, have the property that they are the difference of squares of two consecutive natural numbers?

1. 21
2. 20
3. 22
4. 23

Q. Chandrabhal adds first N natural numbers and finds the sum to be 1850. However, actually one number was added twice by mistake. Find the difference between N and that number.

1. 40
2. 33
3. 60
4. 17

Q. A student of $5^{th}$ standard writing down the counting numbers as 1, 2, 3, 4, ... and then he added all those numbers and got the result 500. But when I checked the result I have found that he had missed a number. What is the missing number?

1. 25
2. 32
3. 30
4. 28

Q. If $1+2+3+\cdots +k=N^2$ and N is less than 100, then the value of k can be (where N $\in$ Natural numbers)

1. 8 and 36
2. both (a) and (b)
3. 8
4. 1 and 49

Q. The integers 1, 2, . . ., 40 are written on a black board. The following operation is then repeated 39 times: In each repetition, any two numbers, say a and b, currently on the blackboard are erased, and a new number a + b - 1 is written. What will be the number left on the board at the ends?

1. 820
2. 821
3. 781
4. 780

Q. Find the ${28383}^{rd}$ digit: 123456789101112....

1. 3
2. 4
3. 9
4. 7

Q. Define a number K such that it is the sum of the squares of the first M natural numbers. (i.e. $K=1^2+2^2+\dots \dots +M^2)$ where M < 55. How many values of M exist such that K is divisible by 4?

1. 12
2. None of these
3. 10
4. 11

Q. A person starts typing the numebrs from 1 to 1999. He press the keys total 'n' number of times. The value of n is:

1. 6889
2. 2888
3. None of these
4. 1000

Q. All the soldiers are arranged in the form of an equilateral triangle i.e., one soldier in the front and 2 soldiers in the second row and 3 soldiers in the third row, 4 soldiers in the fourth row and so on. If 669 more soldiers of another company are added in such a way that all the soldiers now are in the form of a square and each of the sides then contains 8 soldiers less than each side of the equilateral triangle. Initially, how many soldiers were there?

1. 1220
2. 2056
3. 1540
4. 1400