Exam 8: Sequences, Series, and Probability

Use mathematical induction to prove the following for every positive integer $n$ . $\sum _ { i = 1 } ^ { n } \frac { 11 } { i ( i + 1 ) ( i + 2 ) } = \frac { 11 n ( n + 3 ) } { 4 ( n + 1 ) ( n + 2 ) }$

Determine the sample space for the experiment. Two marbles are selected from marbles labeled A through $D$ where the marbles are not replaced and the order of selection does not matter.

Write an expression for the apparent $n$ th term of the sequence. (Assume that $n$ begins with 1.) $- 5 , - 2,1,4,7$

Find the indicated $n$ th term of the geometric sequence. 4th term: $4 , - 12,36 , \ldots$

Prove the inequality for the indicated integer values of $n$ . $\left( \frac { 9 } { 8 } \right) ^ { n } > n , n \geq 29$

Find the sum. $\sum _ { k = 1 } ^ { 3 } \frac { 1 } { k ^ { 2 } + 5 }$

The first two terms of the arithmetic sequence are given. Find the indicated term.

Use mathematical induction to prove the following for every positive integer $n$ . $\sum _ { i = 1 } ^ { n } \frac { 15 } { ( 2 i - 1 ) ( 2 i + 1 ) } = \frac { 15 n } { 2 n + 1 }$

Find the number of distinguishable permutations of the group of letters. $\mathrm { G } , \mathrm { A } , \mathrm { U } , \mathrm { S } , \mathrm { S }$

Find the sum of the integers from $- 1$ to 27 .

Solve for $n$ . $42 \cdot { } _ { n - 1 } P _ { 5 } = { } _ { n + 1 } P _ { 6 }$

Use the Binomial Theorem to expand the complex number. Simplify your result. $( - 2 + 5 i ) ^ { 4 }$

You are given the probability that an event will not happen. Find the probability that the event will happen. $P \left( E ^ { \prime } \right) = \frac { 5 } { 9 }$

Evaluate using a graphing utility: ${ } _ { 15 } P _ { 6 }$

Find the probability for the experiment of tossing a coin four times and getting at least two heads. Use the sample space Sequals HHHH HHHT HHTH HTHH THHH HHTT HTHT THHT THTH HTTH TTHH TTTH TTHT THTT HTTT TTTT

Use the Binomial Theorem to expand and simplify the expression. $( w - 2 ) ^ { 5 }$

Use mathematical induction to prove the following for every positive integer $n$ . $\sum _ { i = 1 } ^ { n } \frac { 5 } { ( 2 i - 1 ) ( 2 i + 1 ) } = \frac { 5 n } { 2 n + 1 }$

Use mathematical induction to prove the following equality. $\ln \left( 2 ^ { n } x _ { 1 } x _ { 2 } \ldots x _ { n } \right) = \ln \left( 2 x _ { 1 } \right) + \ln \left( 2 x _ { 2 } \right) + \ldots + \ln \left( 2 x _ { n } \right)$ , where $x _ { 1 } > 0 , x _ { 2 } > 0 , \ldots , x _ { n } > 0$

Determine whether the sequence is geometric. If so, find the common ratio. $2 , - 2 , - 6 , - 10 , \ldots$

You are given the probability that an event will happen. Find the probability that the event will not happen. $P ( E ) = 0.33$