What is the nth term of the geometric sequence: 3, 6, 12, 24, ...?
3^n
3^(n+1)
2^n
2^(n+1)
Correct Answer :
B. 3^(n+1)
Related Questions
If F(x) is an antiderivative of f(x), what is the relationship between F(b) and F(a) in the context of the Fundamental Theorem of Calculus?
F(b) = F(a)
F(b) - F(a) = f(b) - f(a)
F(b) - F(a) = ?(a to b) f(x) dx
F(b) - F(a) = 0
What does the second derivative f''(x) tell you about a function f(x)?
The rate of change of f(x)
The concavity of f(x)
The slope of the tangent line to f(x)
The area under the curve of f(x)
The Alternating Series Test can be used to determine the convergence of a series if:
The series terms are all positive.
The series terms alternate in sign and decrease in absolute value.
The series has a finite number of terms.
The series has a constant rate of growth.
What is the limit of (sin(x))/x as x approaches 0?
0
1
p
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If a curve C is parameterized by r(t) = , what type of curve is it?
A line
A circle
A parabola
An ellipse
In polar coordinates, what is the equation of a circle with radius r and center at the origin?
r = 1
r = vx^2 + y^2
r = 2?
r = 2
A solution to a first-order linear homogeneous differential equation always has:
Two arbitrary constants
One arbitrary constant
No arbitrary constants
Three arbitrary constants
The Mean Value Theorem for Integrals states that for a continuous function f(x) on the interval [a, b], there exists a c in (a, b) such that:
f(c) = 0
f'(c) = 0
f(c) = (b - a) * f'(c)
f'(c) = 1/(b - a)
What is the limit of (x^2 - 1)/(x - 1) as x approaches 1?
1
0
2
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Separation of variables is a technique commonly used to solve which type of differential equation?
Linear differential equations
Exact differential equations
First-order ordinary differential equations
Second-order ordinary differential equations
The work done by a force F(x) in moving an object along a straight line from x = a to x = b is given by the integral ?(a to b) F(x) dx. What does this integral represent?
The average value of F(x) on the interval [a, b]
The derivative of F(x) with respect to x
The area under the curve of F(x) from a to b
The total work done by the force
What is the Maclaurin series expansion of sin(x)?
x - (x^3)/3! + (x^5)/5! - (x^7)/7! + ...
x + (x^3)/3! + (x^5)/5! + (x^7)/7! + ...
1 - x + (x^2)/2 - (x^3)/3 + ...
e^x
What is the derivative of the function g(x) = e^(3x^2)?
6x * e^(3x^2)
3x^2 * e^(3x^2)
6x * e^(3x)
e^(3x)
What is the integral of 2x dx?
x^2 + C
x^2
x + C
2x^2 + C
If a curve C is parameterized by r(t) = , what type of curve is it?
A line
A circle
A parabola
An ellipse
Which of the following is a method for approximating definite integrals using rectangles?
Trapezoidal Rule
Simpson's Rule
Fundamental Theorem of Calculus
L'H�pital's Rule
If f(x) = x^3 - 12x^2 + 45x, where does f(x) have a local maximum?
x = 3
x = 6
x = 9
x = 12
What is the sum of the first n terms of an arithmetic series with first term a and common difference d?
n(a + d)
n(2a + (n-1)d)/2
n(a + d)/2
a + nd
Which of the following parametric equations represents a horizontal line?
x = t, y = 0
x = 0, y = t
x = t, y = t
x = 0, y = 0
What is the derivative of the vector-valued function r(t) = ?
<2t, 3, 6t^2>
<2t, 3t, 6t^2>
<2t, 3t, 4t^2>
Which test can be used to determine the convergence or divergence of a series when the terms are positive, decreasing, and approach zero?
Integral Test
Ratio Test
Alternating Series Test
Divergence Test
What is the general solution to the differential equation dy/dx + 2y = 4x?
y(x) = 2x^2 + C
y(x) = 2x^2 + 4x + C
y(x) = 2x^2 + 4x
y(x) = 2x^2
Which of the following parametric equations represents a horizontal line?
x = t, y = 0
x = 0, y = t
x = t, y = t
x = 0, y = 0
What is the curl of the vector field F(x, y, z) = <2xy, x^2 - y^2, 3z^2>?
<0, 0, 0>
<0, 2z, 0>
<2, -2, 0>
<0, 0, 6z>
What is the radius of convergence for the power series ?(n=0 to 8) (2^n*x^n/n!)?
1
2
8
0
Which statement is true about the continuity of a function at a point?
The function must be differentiable at that point.
The function must have a removable discontinuity at that point.
The function must have a jump discontinuity at that point.
The function must have the same limit from the left and right at that point.
What is the sum of the first n terms of an arithmetic series with first term a and common difference d?
n(a + d)
n(2a + (n-1)d)/2
n(a + d)/2
a + nd
Which integration technique is used to find the integral of rational functions by decomposing them into partial fractions?
Integration by parts
Trigonometric substitution
Integration by substitution
Partial fraction decomposition
In polar coordinates, what is the equation of a circle with radius r and center at the origin?
r = 1
r = vx^2 + y^2
r = 2?
r = 2
If a function f(x) has a local minimum at x = a, what can be said about f'(a) and f''(a)?
f'(a) = 0 and f''(a) < 0
f'(a) < 0 and f''(a) = 0
f'(a) = 0 and f''(a) = 0
f'(a) > 0 and f''(a) > 0