Answer:
A(w) = - + 100w
Step-by-step explanation:
First, we need to find down the length (l) of the rectangular area in term of w
We have 200 feet of fencing is the perimeter of the rectangular area, which mean:
2 (w + l) = 200
<=> l = 100-w
But the area of land: w*l
= w (100-w)
=- + 100w
So, A(w) = - + 100w
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.