Evaluate sigma from (n-1 to 14) 3n+2
2 answers:
The given series 3n + 2 represents an Arithmetic sequence.
Consider the first few terms of the sequence.
For n = 1, the term is 5
For n = 2, the term is 8
For n = 3, the term is 11
Notice that the difference between the terms is constant. Hence it is an arithmetic sequence. We are to find the sum of first 14 terms of the sequence.
The formula for the sum of AP is:
n = number of terms = 14
a1 = first term = 5
d = common difference = 3
Using the values, we get:
So, the answer to this question is 343
Consider the first few terms of the sequence.
For n = 1, the term is 5
For n = 2, the term is 8
For n = 3, the term is 11
Notice that the difference between the terms is constant. Hence it is an arithmetic sequence. We are to find the sum of first 14 terms of the sequence.
The formula for the sum of AP is:
n = number of terms = 14
a1 = first term = 5
d = common difference = 3
Using the values, we get:
So, the answer to this question is 343
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Answer:
The three angles of the triangle are 90, 35.67 and 54.33 degrees.
Step-by-step explanation:
One way to find the angles of the triangle with vertices (1, 1, 1), (1, -3, 2), (-3, 2, 5) is using the definition of the <em>dot product</em> of two vectors, defined as:
a . b = |a| |b| cosФ [1]
Where |a| and |b| are the norms of vectors <em>a</em> and b, and cosФ is the cosine of the angle between either vector <em>a</em> and <em>b</em>.
If a = [ ] and b = [
], then the dot product is simply a number (not a vector) obtained from:
a . b =
The norm of a vector (its length) is, for instance, |a| = , for a vector in
.
Having all that into account, we can determine the angles of the triangle for each vertex using equation [1] and solving it for Ф.
<h3>Angle of the triangle for vertex in (1, 1, 1)</h3>The vectors which form an angle from this vertex are the result of subtracting the vertex (1, 1, 1) to any of the remaining points (1, -3, 2) and (-3, 2, 5):
v(1, 1, 1) - v(1, -3, 2) =
v(1, 1, 1) - v(-3, 2, 5) =
The <em>dot product</em> for these vectors is:
[0, 4, -1] . [4, -1, -4] = [0 * 4 + 4 * -1 + -1 * -4] = 0 - 4 + 4 = 0
The norm for each vector is:
|(0, 4, -1)| =
|(4, -1, -4)| =
So
a . b = |a| |b| cosФ
In vertex (1, 1, 1) the angle of the triangle is 90 degrees. We have here a right triangle.
We have to follow the same procedure for finding the vectors for angles in vertices (1, -3, 2) and (-3, 2, 5), or better, after finding one of the previous angles, we find the remaining angle subtracting the sum of two angles from 180 degrees to finally obtaining the three angles in question.
Therefore, the other angles are 35.67 degrees and 180 - (90 + 35.67) = 180 - 125.67 = 54.33 degrees.
Given :
Marisa wants to buy a home in Atlanta with a 30-year mortgage that has an annual interest rate of 4.9%.
The house she wants is $250,000 and she will make a $55,000 down payment and borrow the remainder.
To Find :
What is Marisa's monthly mortgage payment to the nearest dollar.
Solution :
Money remains, m = $( 250000 - 55000 ) = $195000 .
Total price after interest, T = m( 1 + rt )
T = 195000×( 1 + 0.049×1)
T = $204555
Monthly payment,
Therefore, monthly payment is $17046.25 .
Hence, this is the required solution.