express [3/7]+[4/8]+[5/9]+[6/10] in sigma notion.
sigma from x = 3 to x = 6 of [ x/(x+4) ]
how did you find that equation ? it there a general formula for that ?
Nope, just look at it
on top
3,4,5, 6
=2, 3+1 etc
on bottom
7,8,9,10
= 7, 7+1 etc
To express the given expression [3/7]+[4/8]+[5/9]+[6/10] in sigma notation, we need to find a pattern in the terms and the corresponding variable to represent that pattern.
First, let's simplify the fractions:
3/7 = 6/14
4/8 = 1/2
5/9 = 10/18
6/10 = 3/5
Now, let's write the expression in terms of these simplified fractions:
[6/14] + [1/2] + [10/18] + [3/5]
To find the common denominator of these fractions, we can use the least common multiple (LCM) of the denominators, which is 14.
The expression can be rewritten as:
[6/14] + [7/14] + [10/14] + [8/14]
Next, let's observe the pattern in the numerators:
6 + 7 + 10 + 8
We can see that the numerators increase by 1 in each term.
Now, let's write this pattern in sigma notation. We'll use the variable k as the index:
∑ (k+5)/14
The Greek letter sigma (∑) represents the summation, and k is the index variable that starts from 0 (because the first term in our expression is [6/14]). The expression (k+5)/14 represents the pattern in the numerators, where k starts from 0 and increases by 1 in each term.
Therefore, the expression [3/7]+[4/8]+[5/9]+[6/10] can be written in sigma notation as:
∑ (k+5)/14, where the summation is taken from k=0 to 3.
Note that the limits of the summation are determined by the number of terms in the original expression. In this case, we have 4 terms, so the summation goes from k=0 to 3.