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

Use mathematical induction to prove that the statement is true for every positive integer n. - $2 ^ { n } > 2 n + 2 , \text { for } n \geq 4$

Provide an appropriate response. -Consider the sequence defined by a $a _ { n } = - 9 n + 59$ Is this sequence arithmetic, geometric, or neither?

Write the binomial expansion of the expression. - $\left( \frac { 1 } { x } - \sqrt { 7 } y \right) ^ { 3 }$

Use the sequence feature of a graphing calculator to evaluate the sum of the first 10 terms of the arithmetic sequence. Round to the nearest thousandth, if necessary. - $a _ { n } = \sqrt { 5 } n + \sqrt { 3 }$

Find the eccentricity $e$ of the hyperbola. - $16 x ^ { 2 } - y ^ { 2 } = 1$

Write an equation for the ellipse. -foci at $( \pm 2 \sqrt { 10 } , 0 ) ; x$ -intercepts $( \pm 7,0 )$

Use a calculator to evaluate the expression. - ${ } _ { 21 } \mathrm { C } _ { 4 }$

The distribution of B.A. degrees conferred by a local college is listed below, by major. Major Frequency English 2073 Mathematics 2164 Chemistry 318 Physics 856 Liberal Arts 1358 Business 1676 Engineering 868 9313 What is the probability that a randomly selected degree is not in Mathematics?

Fill in the blanks to complete the statement. The graph of $\mathrm { P } ( \mathrm { x } ) = - 0.4 ( \mathrm { x } - 7 ) ^ { 2 }$ is a parabola with ــــــــــ $x$ -intercept(s). Its vertex is theــــــــــ point of the graph and has coordinatesــــــــــــــــ

Write an equation for the parabola. -vertex: $( - 4,1 )$ , directrix: $x = - 9$

Use the summation properties to evaluate the series. The following rules may be needed: $\sum _ { i = 1 } ^ { n } i = \frac { n ( n + 1 ) } { 2 } ; \quad \sum _ { i = 1 } ^ { n } i ^ { 2 } = \frac { n ( n + 1 ) ( 2 n + 1 ) } { 6 } ; \quad \sum _ { i = 1 } ^ { n } i ^ { 3 } = \frac { n ^ { 2 } ( n + 1 ) ^ { 2 } } { 4 }$ - $\sum _ { i = 1 } ^ { 6 } \left( i + i ^ { 2 } \right)$

Find the first six terms of the sequence. - $a _ { 1 } = 9 , a _ { n } = a _ { n - 1 } + 6$

Choose the equation that matches the graph. -

Write an equation for the hyperbola. -vertices at $( 4,0 ) , ( - 4,0 )$ ; foci at $( 5,0 ) , ( - 5,0 )$

Find the common difference for the arithmetic sequence. - $- 2 , - 3 , - 4 , - 5 , \ldots$

Decide whether the given sequence is finite or infinite. - $\mathrm { a } _ { 1 } = 4 , \mathrm { a } _ { 2 } = 8 ; \text { for } \mathrm { n } \geq 3 , \mathrm { a } _ { \mathrm { n } } = 5 \cdot \mathrm { a } _ { \mathrm { n } - 1 } + 8 \cdot \mathrm { a } _ { \mathrm { n } - 2 }$

Suppose that a family has 5 children and that the probability of having a girl is $\frac { 1 } { 2 }$ . What is the probability of having exactly three girls and two boys?

Decide whether the given sequence is finite or infinite. - $- 9 , - 8 , - 7 , - 6 , \ldots$

Use the sequence feature of a graphing calculator to evaluate the sum of the first 10 terms of the arithmetic sequence. Round to the nearest thousandth, if necessary. - $a _ { n } = 3.9 n + 8.96$

Find the probability. -A 6-sided die is rolled. What is the probability of rolling a number less than 3?