Question

the sum of the first 2007 term of the sequence1,2,3,4,1,2,3,4, is

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...  

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In blue: Triangular numbers | Image source mathisfun.com

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 .

Step-by-step explanation: