Exam 14: Sequences, Series, and the Binomial Theorem

Solve the problem. -A pendulum swings through an arc of length 90 inches on its first swing. On each successive swing, the length of the arc is the length of the arc on the preceding swing. Find the length of the Arc on the fourth swing, and find the total arc length for the first four swings. Round to two Decimal places.

Solve the problem. -A ping-pong ball is dropped from a height of $9 \mathrm { ft }$ and always rebounds $\frac { 1 } { 3 }$ of the distance fallen. Find the total sum of the rebound heights of the ball.

Write the first five terms of the sequence whose general term is given. - $a _ { n } = 4 n$

Solve the problem. -A gambler has a specific betting system where the first bet made is $\$ 10$ , the second bet made is $\$ 20$ , the third bet is $\$ 40$ , and so on. Find how much the gambler bets on the seventh bet. Find the total amount of money the gambler has bet over the first seven bets.

The notation $\left( \frac { \mathbf { n } } { \mathrm { r } } \right) \text { means } \left( \frac { \mathrm { n } ! } { \mathrm { r } ! ( \mathrm { n } - \mathrm { r } ) ! } \right) \text {. Evaluate the expression. }$ - $\left( \begin{array} { l } 9 \\ 9 \end{array} \right)$

Find the indicated term. -The third term of the expansion of $( 2 x + 3 ) ^ { 3 }$

Find the indicated term of the sequence. -If the second term of a geometric progression is $\frac { 14 } { 5 }$ and the third term is $- \frac { 28 } { 5 }$ , find a 1 and $\mathrm { r }$ .

Evaluate the expression. - $\frac { 4 ! } { 3 ! }$

Use the partial sum formula to find the partial sum of the given arithmetic sequence. -Find the sum of the first five terms of the arithmetic sequence $9,15,21 , \ldots$

Evaluate the expression. - $\sum _ { i = 2 } ^ { 5 } 6 i$

Find the indicated term for the sequence whose general term is given. - $a _ { n } = 4 n - 1 ; a _ { 5 }$

Find the partial sum of the given sequence. - $\mathrm { S } _ { \infty }$ of the sequence $4 , - \frac { 4 } { 3 } , \frac { 4 } { 9 } , \cdots$

The notation $\left( \frac { \mathbf { n } } { \mathrm { r } } \right) \text { means } \left( \frac { \mathrm { n } ! } { \mathrm { r } ! ( \mathrm { n } - \mathrm { r } ) ! } \right) \text {. Evaluate the expression. }$ - $\left( \begin{array} { c } 10 \\ 5 \end{array} \right)$

Solve the problem. -A bacteria culture starts with 7 units and triples every day. Write the general term of the sequence that describes the growth of this culture. Find the number of bacteria units there will be at the beginning of the fourth day.

Write the first five terms of the sequence whose general term is given. - $a _ { n } = n - 5$

Evaluate the expression. - $\sum _ { i = 1 } ^ { 4 } 3 ^ { i }$

Write the first five terms of the geometric sequence whose first term, a1, and common ratio, r, are given. - $a _ { 1 } = 6 ; r = \frac { 1 } { 4 }$

Write the series with summation notation. - $35 + 7 + \frac { 7 } { 5 } + \frac { 7 } { 25 }$

Solve the problem. -A pendulum swings a length of 20 centimeters on its first swing. Each successive swing is $\frac { 3 } { 4 }$ of the preceeding swing. Find the length of the fourth swing and the total length swung during the first four swings. Round to the nearest tenth of a centimeter, if necessary.

Find the indicated term of the sequence. -The fourth term of the geometric sequence whose first term is 6 and whose common ratio is 2