SOLUTION: the value of 4 + 8 + 12 + 16 + 20 + ...... + 100 is

Algebra ->  Sequences-and-series -> SOLUTION: the value of 4 + 8 + 12 + 16 + 20 + ...... + 100 is      Log On


   

Question 233191: the value of 4 + 8 + 12 + 16 + 20 + ...... + 100 is
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
4 + 8 + 12 + 16 + 20 + ...... + 100

This is an arithmetic series because the difference between
any term (after the first) and the term that precedes it is
4.  So the common difference is given by d+=+4.

The formulas for an arithmetic series are

1.  For the nth term:

a%5Bn%5D+=+a%5B1%5D%2B%28n-1%29d

where a%5B1%5D is the first term

2.  For the sum of the terms through the nth term:

S%5Bn%5D=%28n%28a%5B1%5D%2Ba%5Bn%5D%29%29%2F2

In your problem, a%5B1%5D=4, a%5Bn%5D=100 and d=4, so

we use the first formula to find n:

a%5Bn%5D+=+a%5B1%5D%2B%28n-1%29%2Ad
100+=+4%2B%28n-1%29%2A4
100+=+4%2B4%28n-1%29
100+=+4%2B4n-4
100+=+4n
25=n

So a%5Bn%5D=a%5B25%5D=100.

Then we substitute in the second formula:

S%5Bn%5D=%28n%28a%5B1%5D%2Ba%5Bn%5D%29%29%2F2
S%5B25%5D=%2825%284%2Ba%5B25%5D%29%29%2F2
S%5B25%5D=%2825%284%2B100%29%29%2F2
S%5B25%5D=%2825%28104%29%29%2F2
S%5B25%5D=%2825%28cross%28104%29%5E52%29%29%2Fcross%282%29
S%5B25%5D=25%2A52
S%5B25%5D=1300

---------------------------

Here is another way to do it without using formulas:

S = 4 + 8 + 12 + 16 + 20 + ...... + 100 

S = 4(1 + 2 + 3 + 4 + 5 + ...... + 25) 

so there are 25 terms:

Write the sum, S, forward, then backwards

S =   4 +  8 + 12 + 16 + 20 + ...... + 100
S = 100 + 96 + 92 + 88 + 84 + ...... +   4

Add the two equations term by terms:

 S =   4 +   8 +  12 +  16 +  20 + ...... + 100
 S = 100 +  96 +  92 +  88 +  84 + ...... +   4
-----------------------------------------------
2S = 104 + 104 + 104 + 104 + 104 + ...... + 104

Since there are 25 terms, there are 25 104's above, so

2S = 25*104
2S = 2600
 S = 1300

Your teacher probably wants you to do it the first way, though.

Edwin