Exam 11: Further Topics in Algebra

It can be shown that $( 1 + x ) ^ { n } = 1 + n x + \frac { n ( n - 1 ) } { 2 ! } x ^ { 2 } + \frac { n ( n - 1 ) ( n - 2 ) } { 3 ! } x ^ { 3 } \ldots$ is true for any real number n (not just positive integer values) and any real number x, where $| x | < 1$ Use this series to approximate the given number to the nearest thousandth. - $1 \cdot 3 + 2 \cdot 4 + 3 \cdot 5 + \ldots + ( n + 3 ) ( n + 5 ) = \frac { n \left( 2 n ^ { 2 } + 27 n + 115 \right) } { 6 }$

Decide whether the given sequence is finite or infinite. --3, -2, -1, 0, . . .

Solve the problem. -A man borrowed $2000 at 2.5% interest compounded annually. If he paid off the loan in full at the end of 6 years, how much did he pay?

Solve the problem. -A fair spinner has regions numbered 1 through 21. What is the probability that the spinner will stop on an even number or a multiple of 3?

Find the first term and the common ratio for the geometric sequence. Round approximations to the nearest hundredth. - $a _ { 2 } = - 9 , a _ { 5 } = 243$

Solve the problem. -Find the sum of the first 992 positive even integers.

Solve the problem. -An ordinary die is tossed. What are the odds in favor of the die showing a number less than 3?

How many terms are in the expansion of $( x + y ) ^ { 6 } ?$

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

Solve the problem. -A student is told to work any 7 out of 10 questions on an exam. In how many different ways can he complete the exam? (The correctness of his answers has no bearing.)

Find the first six terms of the sequence. - $a _ { 1 } = 3 , a _ { 2 } = 3 ;$ for $n \geq 3 , a _ { n } = a _ { n - 1 } + a _ { n - 2 }$