Exam 11: Further Topics in Algebra

Use mathematical induction to prove that the statement is true for every positive integer n. -The series of sketches below starts with an equilateral triangle having sides of length 1 (one). In the following steps, equilateral triangles are constructed by joining the midpoints of the sides of the preceding triangle. Develop a formula for the length of the sides of the new triangle.

Use a calculator to evaluate the expression. - ${ } _ { 29 } \mathrm { C } _ { 5 }$

Use mathematical induction to prove that the statement is true for every positive integer n. - $3 ^ { n } > 1$

Find a general term an for the geometric sequence. - $\mathrm { a } _ { 1 } = \frac { 1 } { 2 } , \mathrm { r } = 2$

Find the future value of the annuity. -A sequence of yearly payments of $5000 is invested at the end of each of 9 years at an interest rate of 2.1%, compounded annually. What is the total amount of the annuity after 9 years?

Evaluate the expression. - $\left( \begin{array} { l } 10 \\ 10 \end{array} \right)$

Provide an appropriate response. -Consider the selection of officers for a club. Is this a combination, a permutation, or neither?

Use a calculator to evaluate the expression. - ${ } _ { 32 } \mathrm { P } _ { 4 }$

Solve the problem. -A musician plans to perform 5 selections for a concert. If he can choose from 7 different selections, how many ways can he arrange his program?

Write the binomial expansion of the expression. - $\left( \frac { 1 } { x } - \sqrt { 15 } y \right) ^ { 3 }$

Write the indicated term of the binomial expansion. - $( 3 x - 2 y ) ^{11 }; 9$ th term

Evaluate the expression. -P(12, 12)

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

Evaluate the expression. - ${ } _ { 10 } \mathrm { C } _ { 8 }$

Write the binomial expansion of the expression. - $( 2 x + 4 ) ^ { 5 }$

Solve the problem. -Find the sum of the first 78 positive multiples of 4.

Solve the problem. -What are the odds in favor of spinning a D on this spinner?

Use mathematical induction to prove that the statement is true for every positive integer n. - $9 n > n$

Find all natural number values for n for which the given statement is false. - $n ! > 10 n$

Solve the problem. -Martin saves $5 on the first day of a 30-day period, $10 on the second day, and so on. For the next 30 days, he increases the amount saved by $10 each day (instead of $5 each day). How much will He have saved after 60 days?