Exam 14: Sequences, Series, and the Binomial Theorem

The notation $\left( \frac { \mathbf { n } } { \mathrm { r } } \right) \text { means } \left( \frac { \mathrm { n } ! } { \mathrm { r } ! ( \mathrm { n } - \mathrm { r } ) ! } \right) \text {. Evaluate the expression. }$ - $\left( \begin{array} { l } 7 \\ 0 \end{array} \right)$

Solve the problem. -A stocker at a grocery store has created a display of stacked cans such that the top row contains 2 cans, the second row contains 4 cans, the third row contains 6 cans, and so on for six rows. Write the finite series of this sequence, and find the total number of cans in the display.

Evaluate the expression. - $\frac { 4 ! } { 6 ! }$

Fill in the blank with one of the words or phrases listed below. general term common difference finite sequence common ratio Pascal's triangle infinite sequence factorial of series geometric sequence arithmetic sequence -The --------, written n!, is the product of the first n consecutive natural numbers.

Write the first five terms of the sequence whose general term is given. - $a _ { n } = ( - 1 ) ^ { n } ( n + 3 )$

Use the binomial formula to expand the binomial. - $( \mathrm { w } - \mathrm { s } ) ^ { 6 }$

Find the indicated term(s) of the given sequence. -The eighth term of the sequence $a _ { n } = 7 - 2 ( n - 1 )$

Solve the problem. -A particular substance decays in such a way that it loses half its weight each day. How much of the substance is left after 7 days if it starts out at 32 grams?

Solve the problem. -To train for a race, Will begins by jogging 13 minutes one day per week. He increases his jogging time by 6 minutes each week. Write the general term of this arithmetic sequence, and find how many weeks it takes for him to reach a jogging time of one hour.

Find the indicated term(s) of the given sequence. -The general term of the sequence $3 , \frac { 3 } { 2 } , \frac { 3 } { 4 } , \frac { 3 } { 8 } , \frac { 3 } { 16 } , \ldots$

Find the indicated term of the sequence. -The twentieth term of the arithmetic sequence $13,16,19 , \ldots$

Solve the problem. -A job pays a salary of 33,000 the first year. During the next 8 years, the salary increases by 3% each year. What is the salary for the 9th year? What is the total salary over the 9-year period? Round to The nearest cent.

Given are the first three terms of a sequence that is arithmetic. Find a1 and d. - $9,13,17$

Use Pascal's triangle to expand the binomial. - $( a + b ) ^ { 4 }$

Use the partial sum formula to find the partial sum of the given geometric sequence. -Find the sum of the first six terms of the geometric sequence $3,15,75 , \ldots$

Solve the problem. -After being struck with a hammer, a gong vibrates 32 vibrations in the first second and in each second thereafter makes $\frac { 4 } { 5 }$ as many vibrations as in the previous second. Find how many vibrations the gong makes before it stops vibrating.

Write the series with summation notation. - $6 + 18 + 54 + 162 + 486$

Use the binomial formula to expand the binomial. - $( x - 1 ) ^ { 6 }$

Find the indicated term for the sequence whose general term is given. - $a _ { n } = \frac { ( - 1 ) ^ { n } } { n + 8 } ; a _ { 19 }$

Find the sum of the terms of the infinite geometric sequence. - $1 , \frac { 1 } { 2 } , \frac { 1 } { 4 } , \cdots$