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Dayaan has provided examples of their subject expertise by answering 93 Math questions submitted by students on Wyzant’s Ask an Expert.
QUESTION
Dayaan M.’s ANSWER
To write each series in sigma notation, we need to find the pattern in the terms. Part a:For pa...
QUESTION
Dayaan M.’s ANSWER
QUESTION
Dayaan M.’s ANSWER
The difference between the two approaches comes down to what you are allowed to use as evidence. In the geometry you probably did first, which i...
Melvin B.
asked • 10/19/19
answered • 07/10/26
To write each series in sigma notation, we need to find the pattern in the terms.
Part a:
For part a, the series is:
3 + 2 + 5/3 + 3/2
Firstly, we can rewrite the whole numbers as fractions so the pattern is easier to recognize.
3 can be written as 3/1.
2 can be written as 4/2.
So, the series becomes:
3/1 + 4/2 + 5/3 + 6/4
Now, we can see the pattern. The denominators are 1, 2, 3, and 4. That means the denominator can be represented by n. The numerators are 3, 4, 5, 6. Each numerator is 2 more than the denominator. So, the numerator can be represented by n + 2. Therefore, the rule for each term is:
(n + 2) / n
Since there are 4 terms, n starts at 1 and ends at 4. So, for part a, it would be:
∑[(n + 2) / n] for n = 1 to n = 4
Part b:
For part b, the series is:
5 + 8 + 11 + 14 + 17 + 20
Firstly, look at how the terms change. From 5 to 8, 3 is being added. From 8 to 11, another 3 is being added and so on the pattern repeats. Since the same number is being added each time, this is an arithmetic sequence.
For an arithmetic sequence, the rule is:
an = a1 + (n - 1)d
where,
an = term you are finding
a1 = first term
n = term's position
d = common difference
So, we can substitute the first term and common difference:
5 + 3(n - 1)
Since there are 6 terms, n starts at 1 and ends at 6. So, the summation for part b would be:
∑[5 + 3(n - 1)] for n = 1 to n = 6
Tetro T.
asked • 10/15/21
answered • 02/10/26
none n.
asked • 05/28/16
answered • 1d
The difference between the two approaches comes down to what you are allowed to use as evidence. In the geometry you probably did first, which is usually called synthetic or Euclidean geometry, you work from the picture and from a list of postulates and theorems. If you want to prove two triangles are congruent, you hunt for matching sides and angles and you cite something like SAS. There are no numbers attached to the points at all.
In analytic geometry you drop the figure onto a coordinate grid first, and after that every geometric statement turns into an algebra problem. Two lines being perpendicular stops being a picture fact and becomes the statement that their slopes multiply to −1. The distance between two points stops being something you measure and becomes
d = √((x2 − x1)2 + (y2 − y1)2)
which is really just the Pythagorean theorem wearing coordinates. A circle stops being the set of points a fixed distance from a center and becomes (x − h)2 + (y − k)2 = r2. That is what Descartes gave us, a translation dictionary between shapes and equations.
