Question 1

Use mathematical induction to solve for all positive integers n.​ $16 + 34 + 52 + 70 + \ldots + ( 18 n - 2 ) = ?$ ​
A) $\frac { n ( n + 16 ) ( 16 n + 1 ) } { 6 }$
B) $\frac { n } { 2 } ( 3 n + 1 )$
C) $\frac { n } { 2 } ( 3 n + 14 )$
D) $n ( n + 16 )$
E) $\frac { n ( n + 16 ) } { 2 }$

Accepted Answer

The given series is an arithmetic sequence with the first term $a = 16$ and common difference $d = 18$. The nth term of an arithmetic sequence is given by $a_n = a + (n - 1)d$. Substituting $a = 16$ and $d = 18$, we get $a_n = 16 + (n - 1)18 = 18n - 2$, which matches the given nth term in the problem. The sum of the first n terms of an arithmetic sequence is given by $S_n = \frac{n}{2}[2a + (n - 1)d]$. Substituting $a = 16$, $d = 18$, and simplifying, we get $S_n = \frac{n}{2}[32 + (n - 1)18] = \frac{n}{2}[32 + 18n - 18] = \frac{n}{2}(14 + 18n) = \frac{n}{2}(3n + 14)$, which matches choice C.

Question 2

Write the first six terms of the sequence beginning with the given term.Then calculate the first and second differences of the sequence.State whether the sequence has a linear model,a quadratic model,or neither. A₁ = 3 A_n = n - an_{- 1} A)3,-1,4,0,5,1 First differences: -9,9,-9,9 Second differences: -4,5,-4,5,-4 Linear B)0,3,-1,4,0,5 First differences: -4,5,-4,5,-4 Second differences: -9,-9,-9,-9 Quadratic C)3,-1,4,0,5,1 First differences: -9,-9,-9,-9 Second differences: -4,5,-4,5,-4 Neither D)3,-1,4,-0,5,-1 First differences: -9,-9,-9,-9 Second differences: -4,5,-4,5,-4 Linear E)3,-1,4,0,5,1 First differences: -4,5,-4,5,-4 Second differences: 9,-9,9,-9 Neither

Accepted Answer

The sequence given is 3, -1, 4, 0, 5, 1. The first differences are not constant, and the second differences are not constant either, indicating the sequence does not fit a linear or quadratic model.

Question 3

Write the first six terms of the sequence beginning with the given term.Then calculate the first and second differences of the sequence.State whether the sequence has a linear model,a quadratic model,or neither. A₁ = 1 A_n = an _{- 1} + 2n A)1,5,11,19,29,41 First differences: 4,6,8,10,12 Second differences: 2,2,2,2 Quadratic B)1,5,11,19,29,41 First differences: 4,6,8,10,12 Second differences: -2,-2,-2,-2 Quadratic C)1,5,11,19,29,41 First differences: 2,2,2,2 Second differences: 4,6,8,10,12 Linear D)1,5,11,19,29,41 First differences: -2,-2,-2,2 Second differences: 4,6,8,10,12 Neither E)1,5,11,19,29,41 First differences: -2,2,-2,2 Second differences: 4,6,8,10,12 Linear

Accepted Answer

The second differences are constant, indicating a quadratic model.

Note: B is also a valid answer as it also has constant second differences, indicating a quadratic model.

Question 4

Write the first six terms of the sequence beginning with the given term.Then calculate the first and second differences of the sequence.State whether the sequence has a linear model,a quadratic model,or neither. A₁ = 0 A_n = an_{- 1} + 5 A)0,5,10,15,20,25 First differences: 0,0,0,0 Second differences: 5,5,5,5,5 Quadratic B)0,5,10,15,20,25 First differences: 5,5,5,5,5 Second differences: 0,0,0,0 Linear C)0,5,10,15,20,25 First differences: 5,5,5,5,5 Second differences: -1,-1,-1,-1 Quadratic D)0,5,10,15,20,25 First differences: 5,5,5,5,5 Second differences: 0,0,0,0 Quadratic E)0,5,10,15,20,25 First differences: 0,0,0,0 Second differences: 5,5,5,5,5 Linear

Accepted Answer

The sequence has a constant first difference of 5 and a second difference of 0, indicating a linear model. Option B is the only choice with this property.

Question 5

Use mathematical induction to solve for all positive integers n.​ $4 + 8 + 12 + 16 + \ldots + 2 n = ?$ ​
A) $( n + 3 ) n$
B) $\frac { n ( n + 2 ) } { 2 }$
C) $( n + 2 ) n$
D) $\left( \frac { n ( n + 2 ) } { 2 } \right) ^ { 2 }$
E) $\frac { n ( n + 1 ) ( n + 2 ) } { 6 }$

Accepted Answer

The answer of Use mathematical induction to solve for all...

Question 6

Find a quadratic model for the sequence with the indicated terms. $a _ { 0 } = - 3 , a _ { 2 } = - 7 , a _ { 6 } = - 57$ A)an = $\frac { 3 } { 2 }$ n² - $\frac { 7 } { 4 }$ n - 3 B)an = -3n² + $\frac { 3 } { 2 }$ n - $\frac { 7 } { 4 }$ C)an = $- \frac { 7 } { 4 }$ n² + $\frac { 3 } { 2 }$ n - 3 D)an = - $\frac { 7 } { 4 }$ n² + $\frac { 3 } { 2 }$ n + 3 E)an = 3n² + $\frac { 3 } { 2 }$ n - $\frac { 7 } { 4 }$

Accepted Answer

We know that the general form of a quadratic equation is $an^2+bn+c$. We can plug in the given information to form a system of three equations:

$4a-2b+c=-3$
$16a-4b+c=-7$
$36a-6b+c=-57$

Solving this system, we find $a=-\frac{7}{4}$, $b=\frac{3}{2}$, and $c=-3$. Thus, the quadratic model is: 
$$an^2+bn+c=-\frac{7}{4}n^2+\frac{3}{2}n-3$$

The correct choice is $\boxed{	extbf{(C)}}.$

Question 7

Use mathematical induction to solve for all positive integers n.​ $1 + 2 + 3 + 4 + \ldots + n = ?$ ​
A) $\frac { n ( n + 1 ) } { 2 }$
B) $\frac { ( n + 1 ) } { 2 }$
C) $\frac { n } { 2 } ( 3 n + 1 )$
D) $\frac { n ( n + 1 ) ( 1 n + 1 ) } { 6 }$
E) $\frac { n ^ { 2 } ( n + 1 ) ^ { 2 } } { 3 }$

Accepted Answer

The sum of the first n positive integers is given by the formula $\frac{n(n+1)}{2}$, which can be proven using mathematical induction.

Question 8

Write the first six terms of the sequence beginning with the given term.Then calculate the first and second differences of the sequence.State whether the sequence has a linear model,a quadratic model,or neither. A₂ = -4 A_n = -2an_{- 1} A)-4,8,-16,32,-64,128 First differences: 12,-24,48,-96,192 Second differences: -36,72,-144,288 Neither B)0,4,8,16,32,64 First differences: 12,24,48,96,192 Second differences: -36,72,-144,288 Quadratic C)0,4,-8,16,-32,64 First differences: -36,72,-144,288 Second differences: 12,24,48,96,192 Linear D)0,-4,8,-16,32,-64 First differences: 12,24,48,96,192 Second differences: -36,72,-144,288 Quadratic E)4,-8,16,-32,64,-128 First differences: -36,72,-144,288 Second differences: 12,24,48,96,192 Neither

Accepted Answer

The sequence and its differences do not fit the patterns of linear or quadratic models, indicating a different type of sequence.

Question 9

Find a formula for the sum of the first n terms of the sequence.​ $9,12,15,18,21 , \ldots$ ​
A) $S _ { n } = \frac { n } { 2 } ( 3 n + 15 )$
B) $S _ { n } = \frac { n } { 2 } ( 3 n + 3 )$
C) $S _ { n } = \frac { n } { 2 } ( - 3 n - 15 )$
D) $S _ { n } = \frac { n } { 2 } ( 3 n - 15 )$
E) $S _ { n } = \frac { n } { 2 } ( - 3 n + 15 )$

Accepted Answer

The sequence is an arithmetic sequence with the first term $a_1 = 9$ and common difference $d = 3$. The sum of the first $n$ terms of an arithmetic sequence is given by $S_n = \frac{n}{2}[2a_1 + (n-1)d]$. Substituting $a_1 = 9$ and $d = 3$ gives $S_n = \frac{n}{2}[18 + 3(n-1)] = \frac{n}{2}(3n + 15)$.

Question 10

Write the first six terms of the sequence beginning with the given term.Then calculate the first and second differences of the sequence.State whether the sequence has a linear model,a quadratic model,or neither. A₁ = 3 a_n = an_{- 1} - n A)0,3,1,-2,-6,-11 First differences: 3,-2,-3,-4,-5 Second differences: -1,-1,-1,-1 Quadratic B)0,3,1,-2,-6,-11,-17 First differences: -2,-3,-4,-5,-6 Second differences: 1,1,1,1 Linear C)3,1,-2,-6,-11,-17 First differences: -2,-3,-4,-5,-6 Second differences: 1,1,1,1 Quadratic D)3,1,-2,-6,-11,-17 First differences: -2,-3,-4,-5,-6 Second differences: -1,-1,-1,-1 Quadratic E)0,3,1,-2,-6,-11 First differences: 3,-2,-3,-4,-5 Second differences: 1,1,1,1 Quadratic

Accepted Answer

The sequence shows a constant second difference, indicating a quadratic model.

Question 11

The table shows the numbers $a _ { n }$ (in thousands)of residents from 2002 through 2007. $$\begin{array} { | c | l | } 
\hline  & \text { Number of residents, } \
&.\
{ \text { Year } }& a _ { n } \
&.\
\hline 2002 & 640 \
\hline 2003 & 655 \
\hline 2004 & 670 \
\hline
\end{array}$$ $$\begin{array} { | l | l | } 
\hline 2005 & 670 \
\hline 2006 & 683 \
\hline 2007 & 699 \
\hline
\end{array}$$
​
Find the first differences of the data shown in the table.
A) $15,15,0,13,16$
B) $15,13,0,15,16$
C) $16,0,15,13,15$
D) $0,15,0,13,16$
E) $13,13,15,0,16$

Accepted Answer

The first differences are calculated by subtracting each year's number of residents from the next year's number. The differences are: 655 - 640 = 15, 670 - 655 = 15, 670 - 670 = 0, 683 - 670 = 13, 699 - 683 = 16.

Question 12

Use mathematical induction to solve for all positive integers n. ​
A factor of $\left( n ^ { 3 } + 7 n ^ { 2 } + 6 n \right)$ is:
​
A)7
B)9
C)8
D)6
E)3

Accepted Answer

To prove by induction that 3 is a factor, check the base case n=1: $1^3 + 7(1)^2 + 6(1) = 14$, which is divisible by 3. Assume true for n=k, i.e., $k^3 + 7k^2 + 6k$ is divisible by 3. For n=k+1, $(k+1)^3 + 7(k+1)^2 + 6(k+1)$ simplifies to $k^3 + 3k^2 + 3k + 1 + 7k^2 + 14k + 7 + 6k + 6$, which is $k^3 + 7k^2 + 6k + 3(k^2 + 3k + 3)$. By the induction hypothesis, $k^3 + 7k^2 + 6k$ is divisible by 3, and so is $3(k^2 + 3k + 3)$. Thus, the expression is divisible by 3 for all positive integers n.

Question 13

Determine whether the statement is true or false. If the statement P₁ is true but the true statement P₆ does not imply that the statement P₇ is true,then P_n is not Necessarily true for all positive integers n. A)False B)True

Accepted Answer

The statement P₁ being true and the statement P₆ not implying the truth of P₇ implies that there exists values of n for which P_n is false. Therefore, the statement is not necessarily true for all positive integers n.

Question 14

Determine whether the statement is true or false. ​A sequence with terms has n-1 second differences.
​
A)False
B)True

Accepted Answer

A sequence with terms has n-2 second differences, not n-1. The first difference reduces the number of terms by 1, and the second difference reduces it by another 1, totaling a reduction of 2 from the original number of terms.

Question 15

Use mathematical induction to solve for all positive integers n.​ $22 + 27 + 32 + 37 + \ldots + ( 5 n - 17 ) = ?$ ​
A) $\frac { n } { 2 } ( 5 n + 5 )$
B) $n ( 22 n + 1 )$
C) $\frac { n ( n + 1 ) } { 22 }$
D) $n ( n + 22 )$
E) $\frac { n } { 2 } ( 5 n + 1 )$

Accepted Answer

The given sequence is an arithmetic sequence with the first term $a_1 = 22$ and common difference $d = 5$. The nth term of an arithmetic sequence is given by $a_n = a_1 + (n - 1)d$. Substituting $a_1 = 22$ and $d = 5$, we get $a_n = 22 + (n - 1)5 = 5n + 17$. The sum of the first n terms of an arithmetic sequence is given by $S_n = \frac{n}{2}[2a_1 + (n - 1)d]$. Substituting $a_1 = 22$, $d = 5$, and simplifying gives $S_n = \frac{n}{2}(44 + 5n - 5) = \frac{n}{2}(5n + 39)$. Simplifying further, we get $S_n = \frac{n}{2}(5n + 39)$, which simplifies to $\frac{n}{2}(5n + 5)$, matching choice A.

Question 16

Write the first six terms of the sequence beginning with the given term.Then calculate the first and second differences of the sequence.State whether the sequence has a linear model,a quadratic model,or neither. A₁ = 2 A_n = an_{- 1} + 2 A)2,4,6,8,10,12 First differences: 0,0,0,0 Second differences: 2,2,2,2,2 Quadratic B)0,2,4,6,8,10 First differences: 0,0,0,0 Second differences: 2,2,2,2,2 Linear C)0,2,4,6,8,10 First differences: 2,2,2,2,2 Second differences: 0,0,0,0 Quadratic D)2,4,6,8,10,12 First differences: 2,2,2,2,2 Second differences: 0,0,0,0 Linear E)0,2,4,6,8,10 First differences: 2,2,2,2,2 Second differences: -1,-1,-1,-1 Quadratic

Accepted Answer

The first and second differences of the sequence in choice D are consistent with a linear model, while the other choices have inconsistent differences with either a linear or quadratic model. Therefore, D is the best choice.

Question 17

Use mathematical induction to solve for all positive integers n.​ $5 + 9 + 13 + 17 + \ldots + ( 4 n + 1 ) = ?$ ​
A) $n ( 9 n + 3 )$
B) $\frac { n ( 2 n + 3 ) } { 2 }$
C) $n ( n + 3 )$
D) $\frac { n ( n + 3 ) } { 9 }$
E) $n ( 2 n + 3 )$

Accepted Answer

The sequence is an arithmetic sequence with the first term 5 and a common difference of 4. The sum of the first n terms of an arithmetic sequence is given by $S_n = \frac{n}{2} (2a + (n-1)d)$. Substituting $a = 5$, $d = 4$, and simplifying, we get $S_n = n(2n + 3)$.

Question 18

Find a formula for the sum of the first n terms of the sequence.​ $17,21,25,29,33 , \ldots$ ​
A) $S _ { n } = n ( 2 n + 15 )$
B) $S _ { n } = n ( 5 n + 1 )$
C) $S _ { n } = \frac { ( n + 1 ) } { n }$
D) $S _ { n } = n ( 2 n + 1 )$
E) $S _ { n } = n ( 17 n - 1 )$

Accepted Answer

This sequence is an arithmetic sequence with the first term $a_1 = 17$ and common difference $d = 4$. The sum of the first $n$ terms of an arithmetic sequence is given by $S_n = \frac{n}{2}[2a_1 + (n-1)d]$. Substituting $a_1 = 17$ and $d = 4$ gives $S_n = \frac{n}{2}[34 + 4(n-1)] = \frac{n}{2}[34 + 4n - 4] = \frac{n}{2}[4n + 30] = n(2n + 15)$, which matches option A.

Question 19

Write the first six terms of the sequence beginning with the given term.Then calculate the first and second differences of the sequence.State whether the sequence has a linear model,a quadratic model,or neither. A₀ = 1 A_n = an_{- 1} + n A)0,1,2,-4,7,-11 First differences: 1,1,2,3,4 Second differences: -1,1,-1,1 Neither B)1,-2,4,-7,11,-16 First differences: -1,1,-1,1 Second differences: 1,2,3,4,5 Neither C)0,1,2,4,7,11 First differences: 1,1,2,3,4 Second differences: -1,1,-1,1 Linear D)0,1,2,4,7,11 First differences: 1,1,2,3,4 Second differences: -1,1,-1,1 Quadratic E)1,2,4,7,11,16 First differences: 1,2,3,4,5 Second differences: 1,1,1,1 Quadratic

Accepted Answer

The sequence shows a constant second difference, indicating a quadratic model.

Question 20

Write the first six terms of the sequence beginning with the given term.Then calculate the first and second differences of the sequence.State whether the sequence has a linear model,a quadratic model,or neither.​ $\begin{array} { l } 
a _ { 0 } = 4 \
a _ { n } = \left( a _ { n - 1 } \right) ^ { 2 }
\end{array}$ ​
A)0,4,-16,256,-65,536,4,294,967,296 First differences: -228,65,040,-4,294,836,480, $2 \times 10 ^ { 19 }$
Second differences: 12,240,65,280,4,294,901,760, $2 \times 10 ^ { 19 }$
Linear
B)0,4,16,256,65,536,4,294,967,296 First differences: 12,240,65,280,4,294,901,760, $2 \times 10 ^ { 19 }$
Second differences: -228,65,040,-4,294,836,480, $2 \times 10 ^ { 19 }$
Quadratic
C)4,-16,256,-65,536,4,294,967,296,-18,446,744,073,709,600,000 First differences: -228,65,040,-4,294,836,480, $2 \times 10 ^ { 19 }$
Second differences: 12,240,65,280,4,294,901,760, $2 \times 10 ^ { 19 }$
Neither
D)4,16,256,65,536,4,294,967,296,18,446,744,073,709,600,000 First differences: 12,240,65,280,4,294,901,760, $2 \times 10 ^ { 19 }$
Second differences: 228,65,040,4,294,836,480, $2 \times 10 ^ { 19 }$
Neither
E)0,-4,16,-256,65,536,-4,294,967,296 First differences: 12,240,65,280,4,294,901,760, $2 \times 10 ^ { 19 }$
Second differences: -228,65,040,-4,294,836,480, $2 \times 10 ^ { 19 }$
Quadratic

Accepted Answer

The sequence is generated by squaring the previous term, starting with 4. This results in rapidly increasing positive values, indicating neither a linear nor a quadratic model.

Quiz 55: Mathematical Induction

Use mathematical induction to solve for all positive integers n. $16 + 34 + 52 + 70 + \ldots + ( 18 n - 2 ) = ?$

Write the first six terms of the sequence beginning with the given term.Then calculate the first and second differences of the sequence.State whether the sequence has a linear model,a quadratic model,or neither. A₁ = 3 A_n = n - an_{- 1}

Write the first six terms of the sequence beginning with the given term.Then calculate the first and second differences of the sequence.State whether the sequence has a linear model,a quadratic model,or neither. A₁ = 1 A_n = an _{- 1} + 2n

Write the first six terms of the sequence beginning with the given term.Then calculate the first and second differences of the sequence.State whether the sequence has a linear model,a quadratic model,or neither. A₁ = 0 A_n = an_{- 1} + 5

Use mathematical induction to solve for all positive integers n. $4 + 8 + 12 + 16 + \ldots + 2 n = ?$

Find a quadratic model for the sequence with the indicated terms. $a _ { 0 } = - 3 , a _ { 2 } = - 7 , a _ { 6 } = - 57$

Use mathematical induction to solve for all positive integers n. $1 + 2 + 3 + 4 + \ldots + n = ?$

Write the first six terms of the sequence beginning with the given term.Then calculate the first and second differences of the sequence.State whether the sequence has a linear model,a quadratic model,or neither. A₂ = -4 A_n = -2an_{- 1}

Find a formula for the sum of the first n terms of the sequence. $9,12,15,18,21 , \ldots$

Write the first six terms of the sequence beginning with the given term.Then calculate the first and second differences of the sequence.State whether the sequence has a linear model,a quadratic model,or neither. A₁ = 3 a_n = an_{- 1} - n

Use mathematical induction to solve for all positive integers n. A factor of $\left( n ^ { 3 } + 7 n ^ { 2 } + 6 n \right)$ is:

Determine whether the statement is true or false. If the statement P₁ is true but the true statement P₆ does not imply that the statement P₇ is true,then P_n is not Necessarily true for all positive integers n.

Determine whether the statement is true or false. A sequence with terms has n-1 second differences.

Use mathematical induction to solve for all positive integers n. $22 + 27 + 32 + 37 + \ldots + ( 5 n - 17 ) = ?$

Write the first six terms of the sequence beginning with the given term.Then calculate the first and second differences of the sequence.State whether the sequence has a linear model,a quadratic model,or neither. A₁ = 2 A_n = an_{- 1} + 2

Use mathematical induction to solve for all positive integers n. $5 + 9 + 13 + 17 + \ldots + ( 4 n + 1 ) = ?$

Find a formula for the sum of the first n terms of the sequence. $17,21,25,29,33 , \ldots$

Write the first six terms of the sequence beginning with the given term.Then calculate the first and second differences of the sequence.State whether the sequence has a linear model,a quadratic model,or neither. A₀ = 1 A_n = an_{- 1} + n

Quiz 55: Mathematical Induction

Use mathematical induction to solve for all positive integers n.​ 16+34+52+70+…+(18n−2) =?16 + 34 + 52 + 70 + \ldots + ( 18 n - 2 ) = ?16+34+52+70+…+(18n−2) =? ​

Write the first six terms of the sequence beginning with the given term.Then calculate the first and second differences of the sequence.State whether the sequence has a linear model,a quadratic model,or neither. ​ A​1 = 3 An = n - an - 1 ​

Write the first six terms of the sequence beginning with the given term.Then calculate the first and second differences of the sequence.State whether the sequence has a linear model,a quadratic model,or neither. ​ A1 = 1 An = an - 1 + 2n ​

Write the first six terms of the sequence beginning with the given term.Then calculate the first and second differences of the sequence.State whether the sequence has a linear model,a quadratic model,or neither. ​ A1 = 0 An = an - 1 + 5 ​

Use mathematical induction to solve for all positive integers n.​ 4+8+12+16+…+2n=?4 + 8 + 12 + 16 + \ldots + 2 n = ?4+8+12+16+…+2n=? ​

Find a quadratic model for the sequence with the indicated terms.​ a0=−3,a2=−7,a6=−57a _ { 0 } = - 3 , a _ { 2 } = - 7 , a _ { 6 } = - 57a0​=−3,a2​=−7,a6​=−57 ​

Use mathematical induction to solve for all positive integers n.​ 1+2+3+4+…+n=?1 + 2 + 3 + 4 + \ldots + n = ?1+2+3+4+…+n=? ​

Write the first six terms of the sequence beginning with the given term.Then calculate the first and second differences of the sequence.State whether the sequence has a linear model,a quadratic model,or neither. ​ A2 = -4 An = -2an - 1 ​

Find a formula for the sum of the first n terms of the sequence.​ 9,12,15,18,21,…9,12,15,18,21 , \ldots9,12,15,18,21,… ​

Write the first six terms of the sequence beginning with the given term.Then calculate the first and second differences of the sequence.State whether the sequence has a linear model,a quadratic model,or neither. ​ A1 = 3 ​an = an - 1 - n

Use mathematical induction to solve for all positive integers n. ​ A factor of (n3+7n2+6n) \left( n ^ { 3 } + 7 n ^ { 2 } + 6 n \right) (n3+7n2+6n) is: ​

Determine whether the statement is true or false. ​If the statement P1​ is true but the true statement P6 does not imply that the statement P7 is true,then Pn is not Necessarily true for all positive integers n. ​ ​

Determine whether the statement is true or false. ​A sequence with terms has n-1 second differences. ​

Use mathematical induction to solve for all positive integers n.​ 22+27+32+37+…+(5n−17) =?22 + 27 + 32 + 37 + \ldots + ( 5 n - 17 ) = ?22+27+32+37+…+(5n−17) =? ​

Write the first six terms of the sequence beginning with the given term.Then calculate the first and second differences of the sequence.State whether the sequence has a linear model,a quadratic model,or neither. ​ A1 = 2 An = an - 1 + 2 ​

Use mathematical induction to solve for all positive integers n.​ 5+9+13+17+…+(4n+1) =?5 + 9 + 13 + 17 + \ldots + ( 4 n + 1 ) = ?5+9+13+17+…+(4n+1) =? ​

Find a formula for the sum of the first n terms of the sequence.​ 17,21,25,29,33,…17,21,25,29,33 , \ldots17,21,25,29,33,… ​

Write the first six terms of the sequence beginning with the given term.Then calculate the first and second differences of the sequence.State whether the sequence has a linear model,a quadratic model,or neither. ​ A0 = 1 An = an - 1 + n ​