Suppose that you want to test the effects of different dishwashing detergents on dishes that have food stuck to them. To test th
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Segment NO is parallel to the segment KL.
Solution:
Given KLM is a triangle.
MN = NK and MO = OL
It clearly shows that NO is the mid-segment of ΔKLM.
By mid-segment theorem,
<em>The segment connecting two points of the triangle is parallel to the third side and is half of that side.</em>
⇒ NO || KL and
Therefore segment NO is parallel to the segment KL.
Consider the lengths of consecutive 1-2 blocks.
block 1 - 1, 2 - length 2
block 2 - 1, 2, 2 - length 3
block 3 - 1, 2, 2, 2 - length 4
block 4 - 1, 2, 2, 2, 2 - length 5
and so on.
Recall the formula for the sum of consecutive positive integers,
Now,
which means that the 1234th term in the sequence occurs somewhere about 1/5 of the way through the 49th 1-2 block.
In the first 48 blocks, the sequence contains 48 copies of 1 and 1 + 2 + 3 + ... + 47 copies of 2, hence they make up a total of
numbers, and their sum is
This leaves us with the contribution of the first 10 terms in the 49th block, which consist of one 1 and nine 2s with a sum of .
So, the sum of the first 1234 terms in the sequence is 2419.