Question

Please help

Part 1.] Match these items.
1.) The positive integers
2.) An ordering of quantities
3.) 2, 4, 8, 16, . . ., 256
4.) 1, 3, 5, 7, . . .
5.) No first term is available
A.) An example of a finite sequence
B.) A sequence
C.) The reason the numbers . . . -4, -2, 0, 2, 4, . . . are not a sequence
D.) The natural numbers
E.) An example of an infinite sequence

Part 2.] Write the first five terms of the sequence. $a_{n}= n^{2}+2$
A.] 3,6,11,18,27
B.] 3,6,9,12,15
C.] 2,3,6,11,18

Part 3.] Write the first five terms of the sequence. $a_{n}= -1( \frac{1}{3})^{n-1}$
A.] -1,-3,-9,-27,-81
B.] -1,$\frac{-1}{3},\frac{-1}{9},\frac{-1}{27},\frac{-1}{81}$
C.] -1,$\frac{1}{3},\frac{-1}{9},\frac{1}{27},\frac{-1}{81}$

Part 4.] In two or more complete sentences, explain whether the sequence is finite or infinite. Describe the pattern in the sequence if it exists, and if possible find the sixth term. $2a,2a^{2}b,2a^{3}b^{2},2a^{4}b^{3}...$

Answer 1

Part 1
1) The positive integers          ----------->  D.) The natural numbers
2.) An ordering of quantities  -----------> B.) A sequence
3.) 2, 4, 8, 16, . . ., 256            ----------->  A.) An example of a finite sequence
4.) 1, 3, 5, 7, . . .                        ----------->  E.) An example of an infinite sequence
5.) No first term is available   ----------->  C.) The reason the numbers . . . -4, -2, 0, 2, 4, . . . are not a sequence
Part 2
$a_n=n^2+2\\ a_1=1+2=3\\ a_2=4+2=6\\ a_3=9+2=11\\ a_4=16+2=18\\ a_5=25+2=27$
The answers is A.
Part 3
$a_n=-1(\frac{1}{3})^{n-1} \\ a_1=-1(\frac{1}{3})^{1-1}=-1\\ a_2=-1(\frac{1}{3})^{2-1}=-\frac{1}{3}\\ a_3=-1(\frac{1}{3})^{3-1}=--\frac{1}{9}\\ a_4=-1(\frac{1}{3})^{4-1}=--\frac{1}{27}\\ a_5=-1(\frac{1}{3})^{5-1}=--\frac{1}{81}$
The answers is B.
Part 4
The pattern in this sequence is:
$a_n=2a^nb^{n-1}$
Let's list first six terms:
$a_n=2a^nb^{n-1}\\ a_1=2a^1b^0=2a\\ a_2=2a^2b^1=2a^2b\\ a_3=2a^3b^2\\ a_4=2a^4b^3\\ a_5=2a^5b^4\\ a_6=2a^6b^5$
This is a infinite sequence. The patern is obvious and we could find any term within the sequence.