Question

94th term of the following sequence?1,1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,........

Answer 1

Answer:

The answer is 14.

The trick is to find a pattern.

Consider numbering the given sequence as follows:

1 is at position 1

2 is at positions 2, 3

3 is at positions 4, 5, 6

4 is at positions 7, 8, 9, 10

and so on...

Notice that the last positions of the individual values form a sequence.

1,3,6,10,15,21...

Sound familiar? It's the triangular series, where the Nth term is given by:

n(n+1)/2 , where n is the value.

So for the value  n , it last appears at a position  P=n(n+1)/2

For example, 3 will last appear at  3∗(3+1)/2=6

Your question is to find the 100th term right? Then  P=100  and you have to find the value of  n  for which

P=100  .

Thus,

P=n(n+1)/2

⟹100=n(n+1)/2

Solving, we get the value as  n=13.6509716980849

And the nearest integer is  14 .