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. - $\left( 1 - \frac { 1 } { 2 } \right) \left( 1 - \frac { 1 } { 3 } \right) \cdots \left( 1 - \frac { 1 } { n + 1 } \right) = \frac { 1 } { n + 1 }$

Graph the function corresponding to the sequence defined. Use the graph to decide whether the sequence converges or diverges. - $a _ { n } = \frac { n + 77 } { 2 n }$

Graph the hyperbola. - $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 4 } = 1$

The roof of a building is in the shape of the hyperbola $\mathrm { y } ^ { 2 } - \mathrm { x } ^ { 2 } = 50$ , where $\mathrm { x }$ and $\mathrm { y }$ are in meters. Refer to the figu: and determine the height $h$ of the outside walls. $A = 6 \mathrm {~m}$

Identify the equation as a parabola, circle, ellipse, or hyperbola. - $9 x ^ { 2 } + 16 y ^ { 2 } = 144$

Match the equation of a hyperbola with its description. - $\frac { ( x + 2 ) ^ { 2 } } { 64 } - \frac { ( y + 3 ) ^ { 2 } } { 25 } = 1$

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

Evaluate the sum using the given information. - =-2,=-4,=2,=0, and \Delta=0.6;()=-4 \Delta

Find the sum of the first 551 positive integers.

Find a formula for the nth term of the arithmetic sequence shown in the graph. -

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

Evaluate the sum. - $\sum _ { i = 1 } ^ { 5 } ( i + 5 )$

Graph the conic section. $9(x+3)^{2}+16(y-4)^{2}=144$

The hyperbola with equation $\frac { x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 64 } = 1$ 1 opens up and down.

Use mathematical induction to prove that the statement is true for every positive integer n. - $4 \cdot 6 + 5 \cdot 7 + 6 \cdot 8 + \ldots + ( n + 3 ) ( n + 5 ) = \frac { n \left( 2 n ^ { 2 } + 27 n + 115 \right) } { 6 }$

Evaluate the sum using the given information. - $x _ { 1 } = 3 , x _ { 2 } = 1 , x _ { 3 } = 0 , x _ { 4 } = - 4$ , and $x _ { 5 } = - 1$ $\sum _ { i = 1 } ^ { 5 } \left( - 2 x _ { i } + 1 \right)$

If the vertex of the graph of a quadratic function lies in quadrant IV and the graph is concave up, how many x-intercepts does the graph have?

Match the equation of the parabola with the appropriate description. - $y = 3 x ^ { 2 } - 5 x + 4$

Evaluate the series, if it converges. - $- 8 - \frac { 16 } { 3 } - \frac { 32 } { 9 } - \frac { 64 } { 27 } + \ldots$

A town has a population of 10,000 people and is increasing by 10% every year. What will the population be at the end of 5 years?