Exam 7: Arithmetic Sequence: Common Difference and First n Terms

Use the formula for S_n to find the sum of the first five terms of the geometric sequence. - $\frac { 2 } { 3 } , \frac { 8 } { 3 } , \frac { 32 } { 3 } , \frac { 128 } { 3 } , \ldots$

(Multiple Choice)

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Question 281

Provide an appropriate response. -What is true of both the first term and the last term in a binomial expansion?

(Short Answer)

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Question 282

Find the eccentricity $e$ of the ellipse. - $x ^ { 2 } + 4 y ^ { 2 } = 36$

(Multiple Choice)

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Question 283

Use mathematical induction to prove that the statement is true for every positive integer n. - $1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + \ldots + n ( n + 1 ) = \frac { n ( n + 1 ) ( n + 2 ) } { 3 }$

(Essay)

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Question 284

Use mathematical induction to prove that the statement is true for every positive integer n. - $n ! > 3 n , \text { for } n \geq 4$

(Essay)

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Question 285

Write the series using summation notation. - $\frac { 3 } { 1 \cdot 2 } + \frac { 4 } { 2 \cdot 3 } + \frac { 5 } { 3 \cdot 4 } + \frac { 6 } { 4 \cdot 5 } + \frac { 7 } { 5 \cdot 6 }$

(Multiple Choice)

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Question 286

Find the common ratio r for the given infinite geometric sequence. -75, 15, 3, 0.6, 0.12, . . .

(Multiple Choice)

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Question 287

Use mathematical induction to prove that the statement is true for every positive integer n. - $6 ^ { n } > 6 ^ { n - 1 }$

(Essay)

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Question 288

Evaluate the sum. Round to two decimal places, if necessary. - $\sum _ { k = 1 } ^ { 4 } 2 ^ { k }$

(Multiple Choice)

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Question 289

Provide an appropriate response. -Explain in your own words how you can tell from the equation whether the graph is a circle or an ellipse.

(Essay)

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Question 290

Graph the conic section. $25 y^{2}-16 x^{2}=400$ $Graph the conic section. 25 y^{2}-16 x^{2}=400$

(Multiple Choice)

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Question 291

What are the odds in favor of drawing a number greater than 2 from these cards?

(Multiple Choice)

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Question 292

Find the eccentricity $e$ of the hyperbola. - $x ^ { 2 } - 2 y ^ { 2 } = 6$

(Multiple Choice)

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Question 293

Evaluate the expression. - $\left( \begin{array} { l } 7 \\0\end{array} \right)$

(Multiple Choice)

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Question 294

Provide an appropriate response. -Convert the decimal $980980980 \ldots$ to a fraction by writing it as the infinite series $\frac { 980 } { 1000 } + \frac { 980 } { 1000 ^ { 2 } } + \frac { 980 } { 1000 ^ { 3 } } + \ldots$

(Multiple Choice)

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Question 295

Decide whether the given sequence is finite or infinite. - $\mathrm { a } _ { 1 } = 8 ; \text { for } \mathrm { n } \geq 2 , \mathrm { a } _ { \mathrm { n } } = 2 \cdot \mathrm { a } _ { \mathrm { n } - 1 }$

(Multiple Choice)

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Question 296

Graph the conic section. $25 x^{2}-9 y^{2}-250 x-18 y+391=0$ $Graph the conic section. 25 x^{2}-9 y^{2}-250 x-18 y+391=0$

(Multiple Choice)

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Question 297

Evaluate the sum using the given information. - $\mathrm { x } _ { 1 } = - 1 , \mathrm { x } _ { 2 } = 5 , \mathrm { x } _ { 3 } = - 5 , \mathrm { x } _ { 4 } = 4 , \text { and } \Delta \mathrm { x } = - 0.9 ; \mathrm { f } ( \mathrm { x } ) = 3 \mathrm { x } ^ { 2 } - \mathrm { x }$ $\sum _ { i = 1 } ^ { 4 } f \left( x _ { i } \right) \Delta x$

(Multiple Choice)

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Question 298

Find the probability. -A bag contains 8 red marbles, 2 blue marbles, and 5 green marbles. What is the probability of choosing a blue marble when one marble is drawn?

(Multiple Choice)

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Question 299

Find the first six terms of the sequence. - $a _ { 1 } = 4 , a _ { 2 } = 2 ; \text { for } n \geq 3 , a _ { n } = a _ { n - 1 } - a _ { n - 2 }$

(Multiple Choice)

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Question 300

Use the formula for S_n to find the sum of the first five terms of the geometric sequence. - $\frac { 2 } { 3 } , \frac { 8 } { 3 } , \frac { 32 } { 3 } , \frac { 128 } { 3 } , \ldots$

Provide an appropriate response. -What is true of both the first term and the last term in a binomial expansion?

Find the eccentricity $e$ of the ellipse. - $x ^ { 2 } + 4 y ^ { 2 } = 36$

Use mathematical induction to prove that the statement is true for every positive integer n. - $1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + \ldots + n ( n + 1 ) = \frac { n ( n + 1 ) ( n + 2 ) } { 3 }$

Use mathematical induction to prove that the statement is true for every positive integer n. - $n ! > 3 n , \text { for } n \geq 4$

Write the series using summation notation. - $\frac { 3 } { 1 \cdot 2 } + \frac { 4 } { 2 \cdot 3 } + \frac { 5 } { 3 \cdot 4 } + \frac { 6 } { 4 \cdot 5 } + \frac { 7 } { 5 \cdot 6 }$

Find the common ratio r for the given infinite geometric sequence. -75, 15, 3, 0.6, 0.12, . . .

Use mathematical induction to prove that the statement is true for every positive integer n. - $6 ^ { n } > 6 ^ { n - 1 }$

Evaluate the sum. Round to two decimal places, if necessary. - $\sum _ { k = 1 } ^ { 4 } 2 ^ { k }$

Provide an appropriate response. -Explain in your own words how you can tell from the equation whether the graph is a circle or an ellipse.

Graph the conic section. $25 y^{2}-16 x^{2}=400$ $Graph the conic section. 25 y^{2}-16 x^{2}=400$

What are the odds in favor of drawing a number greater than 2 from these cards?

Find the eccentricity $e$ of the hyperbola. - $x ^ { 2 } - 2 y ^ { 2 } = 6$

Evaluate the expression. - $\left( \begin{array} { l } 7 \\0\end{array} \right)$

Provide an appropriate response. -Convert the decimal $980980980 \ldots$ to a fraction by writing it as the infinite series $\frac { 980 } { 1000 } + \frac { 980 } { 1000 ^ { 2 } } + \frac { 980 } { 1000 ^ { 3 } } + \ldots$

Decide whether the given sequence is finite or infinite. - $\mathrm { a } _ { 1 } = 8 ; \text { for } \mathrm { n } \geq 2 , \mathrm { a } _ { \mathrm { n } } = 2 \cdot \mathrm { a } _ { \mathrm { n } - 1 }$

Graph the conic section. $25 x^{2}-9 y^{2}-250 x-18 y+391=0$ $Graph the conic section. 25 x^{2}-9 y^{2}-250 x-18 y+391=0$

Find the probability. -A bag contains 8 red marbles, 2 blue marbles, and 5 green marbles. What is the probability of choosing a blue marble when one marble is drawn?

Find the first six terms of the sequence. - $a _ { 1 } = 4 , a _ { 2 } = 2 ; \text { for } n \geq 3 , a _ { n } = a _ { n - 1 } - a _ { n - 2 }$

Review of Basic Concepts

Equations and Inequalities

Graphs and Functions

Polynomials and Rational Functions

Inverse, Exponential, and Logarithmic Functions

Systems and Matrices

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Exam 7: Arithmetic Sequence: Common Difference and First n Terms

Use the formula for Sn to find the sum of the first five terms of the geometric sequence. - 23,83,323,1283,…\frac { 2 } { 3 } , \frac { 8 } { 3 } , \frac { 32 } { 3 } , \frac { 128 } { 3 } , \ldots32​,38​,332​,3128​,…

Provide an appropriate response. -What is true of both the first term and the last term in a binomial expansion?

Find the eccentricity e e e of the ellipse. - x2+4y2=36x ^ { 2 } + 4 y ^ { 2 } = 36x2+4y2=36

Use mathematical induction to prove that the statement is true for every positive integer n. - 1⋅2+2⋅3+3⋅4+…+n(n+1)=n(n+1)(n+2)31 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + \ldots + n ( n + 1 ) = \frac { n ( n + 1 ) ( n + 2 ) } { 3 }1⋅2+2⋅3+3⋅4+…+n(n+1)=3n(n+1)(n+2)​

Use mathematical induction to prove that the statement is true for every positive integer n. - n!>3n, for n≥4n ! > 3 n , \text { for } n \geq 4n!>3n, for n≥4

Write the series using summation notation. - 31⋅2+42⋅3+53⋅4+64⋅5+75⋅6\frac { 3 } { 1 \cdot 2 } + \frac { 4 } { 2 \cdot 3 } + \frac { 5 } { 3 \cdot 4 } + \frac { 6 } { 4 \cdot 5 } + \frac { 7 } { 5 \cdot 6 }1⋅23​+2⋅34​+3⋅45​+4⋅56​+5⋅67​

Find the common ratio r for the given infinite geometric sequence. -75, 15, 3, 0.6, 0.12, . . .

Use mathematical induction to prove that the statement is true for every positive integer n. - 6n>6n−16 ^ { n } > 6 ^ { n - 1 }6n>6n−1

Evaluate the sum. Round to two decimal places, if necessary. - ∑k=142k\sum _ { k = 1 } ^ { 4 } 2 ^ { k }∑k=14​2k

Provide an appropriate response. -Explain in your own words how you can tell from the equation whether the graph is a circle or an ellipse.

Graph the conic section. 25y2−16x2=40025 y^{2}-16 x^{2}=40025y2−16x2=400

What are the odds in favor of drawing a number greater than 2 from these cards?

Find the eccentricity e e e of the hyperbola. - x2−2y2=6x ^ { 2 } - 2 y ^ { 2 } = 6x2−2y2=6

Evaluate the expression. - (70)\left( \begin{array} { l } 7 \\0\end{array} \right)(70​)

Decide whether the given sequence is finite or infinite. - a1=8; for n≥2,an=2⋅an−1\mathrm { a } _ { 1 } = 8 ; \text { for } \mathrm { n } \geq 2 , \mathrm { a } _ { \mathrm { n } } = 2 \cdot \mathrm { a } _ { \mathrm { n } - 1 }a1​=8; for n≥2,an​=2⋅an−1​

Graph the conic section. 25x2−9y2−250x−18y+391=025 x^{2}-9 y^{2}-250 x-18 y+391=025x2−9y2−250x−18y+391=0

Find the probability. -A bag contains 8 red marbles, 2 blue marbles, and 5 green marbles. What is the probability of choosing a blue marble when one marble is drawn?

Find the first six terms of the sequence. - a1=4,a2=2; for n≥3,an=an−1−an−2a _ { 1 } = 4 , a _ { 2 } = 2 ; \text { for } n \geq 3 , a _ { n } = a _ { n - 1 } - a _ { n - 2 }a1​=4,a2​=2; for n≥3,an​=an−1​−an−2​

Review of Basic Concepts

Equations and Inequalities

Graphs and Functions

Polynomials and Rational Functions

Inverse, Exponential, and Logarithmic Functions

Systems and Matrices

Filters

Use the formula for S_n to find the sum of the first five terms of the geometric sequence. - $\frac { 2 } { 3 } , \frac { 8 } { 3 } , \frac { 32 } { 3 } , \frac { 128 } { 3 } , \ldots$

Find the eccentricity $e$ of the ellipse. - $x ^ { 2 } + 4 y ^ { 2 } = 36$

Use mathematical induction to prove that the statement is true for every positive integer n. - $1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + \ldots + n ( n + 1 ) = \frac { n ( n + 1 ) ( n + 2 ) } { 3 }$

Use mathematical induction to prove that the statement is true for every positive integer n. - $n ! > 3 n , \text { for } n \geq 4$

Write the series using summation notation. - $\frac { 3 } { 1 \cdot 2 } + \frac { 4 } { 2 \cdot 3 } + \frac { 5 } { 3 \cdot 4 } + \frac { 6 } { 4 \cdot 5 } + \frac { 7 } { 5 \cdot 6 }$

Use mathematical induction to prove that the statement is true for every positive integer n. - $6 ^ { n } > 6 ^ { n - 1 }$

Evaluate the sum. Round to two decimal places, if necessary. - $\sum _ { k = 1 } ^ { 4 } 2 ^ { k }$

Graph the conic section. $25 y^{2}-16 x^{2}=400$ $Graph the conic section. 25 y^{2}-16 x^{2}=400$

Find the eccentricity $e$ of the hyperbola. - $x ^ { 2 } - 2 y ^ { 2 } = 6$

Evaluate the expression. - $\left( \begin{array} { l } 7 \\0\end{array} \right)$

Decide whether the given sequence is finite or infinite. - $\mathrm { a } _ { 1 } = 8 ; \text { for } \mathrm { n } \geq 2 , \mathrm { a } _ { \mathrm { n } } = 2 \cdot \mathrm { a } _ { \mathrm { n } - 1 }$

Graph the conic section. $25 x^{2}-9 y^{2}-250 x-18 y+391=0$ $Graph the conic section. 25 x^{2}-9 y^{2}-250 x-18 y+391=0$

Find the first six terms of the sequence. - $a _ { 1 } = 4 , a _ { 2 } = 2 ; \text { for } n \geq 3 , a _ { n } = a _ { n - 1 } - a _ { n - 2 }$