1) find n of the arithmetic sequence A1 = -5, An = 63, d = 2
2) Find the common ratio in the geometric sequence:7, -28, 112 ...
3) the vertex of the graph of y = -99(x-7)^2 - 2 is a minimum value ?
a) True
b) False
4) There is a solution to y = 7 and x^2 = -y^2 + 9.
a) true
b) false
5) There is no solution to
x^2 - y^2 = 3 and y^2 - x^2 = 3.
a) true
b) false
6) solve 15^n = 17
7) solve 8^n less than or equal to 25
1) To find the value of 'n' in the arithmetic sequence A1 = -5, An = 63, and d = 2, we can use the formula for the nth term of an arithmetic sequence:
An = A1 + (n-1)d,
where An is the nth term, A1 is the first term, n is the position of the term, and d is the common difference.
Substituting the given values, we have:
63 = -5 + (n-1)(2).
Simplifying the equation:
63 = -5 + 2n - 2.
Combine like terms:
68 = 2n - 2.
Add 2 to both sides:
70 = 2n.
Divide both sides by 2:
n = 35.
Therefore, the value of 'n' in the given arithmetic sequence is 35.
2) To find the common ratio in the geometric sequence 7, -28, 112 ..., we need to find the ratio between consecutive terms.
The common ratio (r) is found by dividing any term (other than the first term) by its preceding term.
Let's divide the second term (-28) by the first term (7):
r = -28 / 7 = -4.
Therefore, the common ratio in the geometric sequence is -4.
3) To determine if the vertex of the graph of the equation y = -99(x-7)^2 - 2 represents a minimum value, we can look at the coefficient of the x^2 term.
In the given equation, the coefficient of x^2 is -99.
Since the coefficient is negative, the graph is a downward-facing parabola. For a downward-facing parabola, the vertex represents the maximum value, not the minimum value.
Therefore, the statement "the vertex of the graph of y = -99(x-7)^2 - 2 is a minimum value" is false (b)
4) To determine if there is a solution to y=7 and x^2 = -y^2 + 9, we can substitute the value of y in the second equation and solve for x.
Substituting y = 7 into the second equation, we have:
x^2 = -(7)^2 + 9.
Simplifying the equation:
x^2 = -49 + 9,
x^2 = -40.
Since the resulting equation has a negative value on the right side, there is no real solution for x.
Therefore, the statement "There is a solution to y = 7 and x^2 = -y^2 + 9" is false (b).
5) To solve the system of equations x^2 - y^2 = 3 and y^2 - x^2 = 3, we can subtract one equation from the other:
(x^2 - y^2) - (y^2 - x^2) = 3 - 3.
Simplifying the equation:
x^2 - y^2 - y^2 + x^2 = 0.
2x^2 - 2y^2 = 0.
Dividing both sides by 2:
x^2 - y^2 = 0.
Factoring the left side:
(x - y)(x + y) = 0.
Setting each factor equal to zero:
x - y = 0 or x + y = 0.
From the equation x - y = 0, we can solve for x:
x = y.
From the equation x + y = 0, we can solve for y:
y = -x.
Since both equations give the same solution, which is x = y, there is a common solution to the system of equations.
Therefore, the statement "There is no solution to x^2 - y^2 = 3 and y^2 - x^2 = 3" is false (b).
6) To solve the equation 15^n = 17 for 'n', we can take the logarithm of both sides of the equation. The most common logarithm used is the natural logarithm (ln), but you can use any base logarithm:
ln(15^n) = ln(17).
Using the property of logarithms, we can bring the exponent down:
n * ln(15) = ln(17).
Divide both sides by ln(15):
n = ln(17) / ln(15).
Using a calculator, compute the right side to get the approximate value of 'n'.
n ≈ 0.8318.
Therefore, the solution for 'n' is approximately 0.8318.
7) To solve the inequality 8^n ≤ 25, we need to find the value of 'n' that satisfies the condition.
Since 8 is a positive number, we don't have to flip the inequality.
Taking the logarithm of both sides, we can use the natural logarithm:
ln(8^n) ≤ ln(25).
Using the property of logarithms, we can bring the exponent down:
n * ln(8) ≤ ln(25).
Divide both sides by ln(8):
n ≤ ln(25) / ln(8).
Using a calculator, compute the right side to get the approximate value of 'n'.
n ≤ 1.378.
Therefore, the solution for 'n' is n ≤ 1.378.
1) To find "n" in the arithmetic sequence, we can use the formula for the nth term: An = A1 + (n - 1)d, where A1 is the first term, An is the nth term, and d is the common difference.
In this case, A1 = -5, An = 63, and d = 2. Plugging these values into the formula, we get 63 = -5 + (n - 1)2.
Simplifying the equation, we have 63 = -5 + 2n - 2. Combining like terms, we get 68 = 2n - 2. Adding 2 to both sides, we get 70 = 2n. Dividing both sides by 2, we find n = 35.
Therefore, the value of "n" in the arithmetic sequence is 35.
2) To find the common ratio in the geometric sequence, we can divide any term by the previous term.
In this case, if we divide the second term (-28) by the first term (7), we get -28/7 = -4.
Therefore, the common ratio in the given geometric sequence is -4.
3) The vertex of a parabola in the form y = ax^2 + bx + c is given by the coordinates (-b/2a, c - b^2/4a).
In the equation y = -99(x - 7)^2 - 2, we have a = -99, b = 0, and c = -2. Plugging these values into the vertex formula, we get the x-coordinate as -b/2a = 0/(-99 * 2) = 0, and the y-coordinate as c - b^2/4a = -2 - 0/4(-99) = -2.
Since the coefficient of the x^2 term is negative, the parabola opens downwards. The vertex of a downward-opening parabola is the maximum value, not the minimum.
Therefore, the statement "the vertex of the graph of y = -99(x - 7)^2 - 2 is a minimum value" is false.
4) To determine if there is a solution to the equation y = 7 and x^2 = -y^2 + 9, we can substitute the value of y into the second equation and see if we get a valid solution for x.
Substituting y = 7 into the second equation, we have x^2 = -(7)^2 + 9. Simplifying, we get x^2 = -49 + 9. Further simplification gives x^2 = -40.
Since the square of a real number cannot be negative, there is no real solution to the equation x^2 = -40. Therefore, the statement "There is a solution to y = 7 and x^2 = -y^2 + 9" is false.
5) To determine if there is a solution to the system of equations x^2 - y^2 = 3 and y^2 - x^2 = 3, we can simplify the equations and see if the resulting equations are contradictory.
Simplifying the first equation, we get x^2 - y^2 = 3. Rearranging, we have x^2 - 3 = y^2.
Substituting this value of y^2 into the second equation, we get (x^2 - 3) - x^2 = 3. Simplifying, we have -3 = 3.
The equation -3 = 3 is not true, which means there is no solution to the system of equations.
Therefore, the statement "There is no solution to x^2 - y^2 = 3 and y^2 - x^2 = 3" is true.
6) To solve the equation 15^n = 17, we need to find the value of n that makes the equation true.
To isolate the exponent, we can take the logarithm of both sides with the base 15. Using the base change formula, we have log(base 15) 15^n = log(base 15) 17.
Simplifying, we get n = log(base 15) 17.
Using a calculator, we can find that log(base 15) 17 ≈ 1.0673.
Therefore, the value of n that satisfies 15^n = 17 is approximately 1.0673.
7) To solve the inequality 8^n ≤ 25, we need to find the values of n that satisfy the inequality.
Taking the logarithm of both sides with the base 8, we have log(base 8) 8^n ≤ log(base 8) 25.
Since log(base 8) 8^n = n and log(base 8) 25 ≈ 1.3208, we get n ≤ 1.3208.
Therefore, the solution to 8^n ≤ 25 is n ≤ 1.3208.