Calculate the percentage increase in applications from the first year to the second year.

Write down the common ratio of the sequence.

Find an expression for $u_n$.

Find the number of student applications the university expects to receive when $n=11$. Express your answer to the nearest integer.

Write down an expression for $v_n$ .

Calculate the total amount of acceptance fees paid to the university in the first $10$ years.

Find $k$.

State whether, for all $n>k$, the university will have places available for all applicants. Justify your answer.

$\frac{12 669-12 300}{12 300}\times 100$            (M1)

$3\%$            A1

[2 marks]

$1.03$             A1

 
Note: Follow through from part (a).

[1 mark]

$\left(u_n=\right) 12 300\times 1.{03}^{n-1}$             A1

[1 mark]

$\left(u_{11}=\right) 12 300\times 1.{03}^{10}$             (M1)

$16530$             A1

Note: Answer must be to the nearest integer. Do not accept $16500$. 

[2 marks]

$\left(v_n=\right) 10380+600\left(n-1\right)$  OR  $600n+9780$             M1A1

Note: Award M1 for substituting into arithmetic sequence formula, A1 for correct substitution.

[2 marks]

$80\times \frac{10}{2}\left(2\left(10380\right)+9\left(600\right)\right)$              (M1)(M1)

Note: Award (M1) for multiplying by $80$ and (M1) for substitution into sum of arithmetic sequence formula.

$\$10 500 000 \left(\$10 464 000\right)$                A1

[3 marks]

$12 300\times 1.{03}^{n-1}<10 380+600\left(n-1\right)$ or equivalent              (M1)

Note: Award (M1) for equating their expressions from parts (b) and (c).

EITHER

graph showing $y=12 300\times 1.{03}^{n-1}$  and $y=10 380+600\left(n-1\right)$              (M1)

OR

graph showing $y=12 300\times 1.{03}^{n-1}-\left(10 380+600\left(n-1\right)\right)$              (M1)

OR

list of values including, $\left(u_{n=}\right) 17537$  and $\left(v_{n=}\right) 17580$              (M1)

OR

$12.4953\dots$ from graphical method or solving numerical equality              (M1)

Note: Award (M1) for a valid attempt to solve.

THEN

$\left(k=\right)13$                A1

[3 marks]

this will not guarantee enough places.                A1

EITHER

A written statement that $u_n>v_n$, with range of $n$.              R1

Example: “when $n=24$ (or greater), the number of applications will exceed the number of places again” (“$u_n>v_n, n\ge 24$”).

OR

exponential growth will always exceed linear growth              R1

Note: Accept an equivalent sketch. Do not award A1R0.

[2 marks]

The percentage increase proved more difficult than anticipated, with many using an incorrect denominator. Part (b) was accessible with many candidates earning at least three marks. The expression for the general term of the arithmetic sequence was found in part (c). A surprising number of candidates found the acceptance fees paid in the tenth year, rather than the required total acceptance fees found in the first ten years. Most candidates were able to earn one mark for multiplying their value by $80$. In part (e), candidates were usually able to find the first point of intersection of $u_n$ and $v_n$, but did not always realize the answer must be an integer. Part (f) required candidates to support their answer through a comparison of growth rates, or by finding a range of values/single data point where $u_n>v_n$ for $n>k$. Though the answer "$u_n$ is geometric, $v_n$ is arithmetic", inferred some understanding, this was insufficient justification and required a description that went a little bit further. It is recommended that teachers provide opportunities for candidates to explore the more advanced features of the GDC. An inappropriate choice of calculator window made it difficult for candidates to assess and appreciate the behaviour of $u_n$ and $v_n$.

The percentage increase proved more difficult than anticipated, with many using an incorrect denominator. Part (b) was accessible with many candidates earning at least three marks. The expression for the general term of the arithmetic sequence was found in part (c). A surprising number of candidates found the acceptance fees paid in the tenth year, rather than the required total acceptance fees found in the first ten years. Most candidates were able to earn one mark for multiplying their value by $80$. In part (e), candidates were usually able to find the first point of intersection of $u_n$ and $v_n$, but did not always realize the answer must be an integer. Part (f) required candidates to support their answer through a comparison of growth rates, or by finding a range of values/single data point where $u_n>v_n$ for $n>k$. Though the answer "$u_n$ is geometric, $v_n$ is arithmetic", inferred some understanding, this was insufficient justification and required a description that went a little bit further. It is recommended that teachers provide opportunities for candidates to explore the more advanced features of the GDC. An inappropriate choice of calculator window made it difficult for candidates to assess and appreciate the behaviour of $u_n$ and $v_n$.

The percentage increase proved more difficult than anticipated, with many using an incorrect denominator. Part (b) was accessible with many candidates earning at least three marks. The expression for the general term of the arithmetic sequence was found in part (c). A surprising number of candidates found the acceptance fees paid in the tenth year, rather than the required total acceptance fees found in the first ten years. Most candidates were able to earn one mark for multiplying their value by $80$. In part (e), candidates were usually able to find the first point of intersection of $u_n$ and $v_n$, but did not always realize the answer must be an integer. Part (f) required candidates to support their answer through a comparison of growth rates, or by finding a range of values/single data point where $u_n>v_n$ for $n>k$. Though the answer "$u_n$ is geometric, $v_n$ is arithmetic", inferred some understanding, this was insufficient justification and required a description that went a little bit further. It is recommended that teachers provide opportunities for candidates to explore the more advanced features of the GDC. An inappropriate choice of calculator window made it difficult for candidates to assess and appreciate the behaviour of $u_n$ and $v_n$.

The percentage increase proved more difficult than anticipated, with many using an incorrect denominator. Part (b) was accessible with many candidates earning at least three marks. The expression for the general term of the arithmetic sequence was found in part (c). A surprising number of candidates found the acceptance fees paid in the tenth year, rather than the required total acceptance fees found in the first ten years. Most candidates were able to earn one mark for multiplying their value by $80$. In part (e), candidates were usually able to find the first point of intersection of $u_n$ and $v_n$, but did not always realize the answer must be an integer. Part (f) required candidates to support their answer through a comparison of growth rates, or by finding a range of values/single data point where $u_n>v_n$ for $n>k$. Though the answer "$u_n$ is geometric, $v_n$ is arithmetic", inferred some understanding, this was insufficient justification and required a description that went a little bit further. It is recommended that teachers provide opportunities for candidates to explore the more advanced features of the GDC. An inappropriate choice of calculator window made it difficult for candidates to assess and appreciate the behaviour of $u_n$ and $v_n$.

The percentage increase proved more difficult than anticipated, with many using an incorrect denominator. Part (b) was accessible with many candidates earning at least three marks. The expression for the general term of the arithmetic sequence was found in part (c). A surprising number of candidates found the acceptance fees paid in the tenth year, rather than the required total acceptance fees found in the first ten years. Most candidates were able to earn one mark for multiplying their value by $80$. In part (e), candidates were usually able to find the first point of intersection of $u_n$ and $v_n$, but did not always realize the answer must be an integer. Part (f) required candidates to support their answer through a comparison of growth rates, or by finding a range of values/single data point where $u_n>v_n$ for $n>k$. Though the answer "$u_n$ is geometric, $v_n$ is arithmetic", inferred some understanding, this was insufficient justification and required a description that went a little bit further. It is recommended that teachers provide opportunities for candidates to explore the more advanced features of the GDC. An inappropriate choice of calculator window made it difficult for candidates to assess and appreciate the behaviour of $u_n$ and $v_n$.

The percentage increase proved more difficult than anticipated, with many using an incorrect denominator. Part (b) was accessible with many candidates earning at least three marks. The expression for the general term of the arithmetic sequence was found in part (c). A surprising number of candidates found the acceptance fees paid in the tenth year, rather than the required total acceptance fees found in the first ten years. Most candidates were able to earn one mark for multiplying their value by $80$. In part (e), candidates were usually able to find the first point of intersection of $u_n$ and $v_n$, but did not always realize the answer must be an integer. Part (f) required candidates to support their answer through a comparison of growth rates, or by finding a range of values/single data point where $u_n>v_n$ for $n>k$. Though the answer "$u_n$ is geometric, $v_n$ is arithmetic", inferred some understanding, this was insufficient justification and required a description that went a little bit further. It is recommended that teachers provide opportunities for candidates to explore the more advanced features of the GDC. An inappropriate choice of calculator window made it difficult for candidates to assess and appreciate the behaviour of $u_n$ and $v_n$.

The percentage increase proved more difficult than anticipated, with many using an incorrect denominator. Part (b) was accessible with many candidates earning at least three marks. The expression for the general term of the arithmetic sequence was found in part (c). A surprising number of candidates found the acceptance fees paid in the tenth year, rather than the required total acceptance fees found in the first ten years. Most candidates were able to earn one mark for multiplying their value by $80$. In part (e), candidates were usually able to find the first point of intersection of $u_n$ and $v_n$, but did not always realize the answer must be an integer. Part (f) required candidates to support their answer through a comparison of growth rates, or by finding a range of values/single data point where $u_n>v_n$ for $n>k$. Though the answer "$u_n$ is geometric, $v_n$ is arithmetic", inferred some understanding, this was insufficient justification and required a description that went a little bit further. It is recommended that teachers provide opportunities for candidates to explore the more advanced features of the GDC. An inappropriate choice of calculator window made it difficult for candidates to assess and appreciate the behaviour of $u_n$ and $v_n$.

The percentage increase proved more difficult than anticipated, with many using an incorrect denominator. Part (b) was accessible with many candidates earning at least three marks. The expression for the general term of the arithmetic sequence was found in part (c). A surprising number of candidates found the acceptance fees paid in the tenth year, rather than the required total acceptance fees found in the first ten years. Most candidates were able to earn one mark for multiplying their value by $80$. In part (e), candidates were usually able to find the first point of intersection of $u_n$ and $v_n$, but did not always realize the answer must be an integer. Part (f) required candidates to support their answer through a comparison of growth rates, or by finding a range of values/single data point where $u_n>v_n$ for $n>k$. Though the answer "$u_n$ is geometric, $v_n$ is arithmetic", inferred some understanding, this was insufficient justification and required a description that went a little bit further. It is recommended that teachers provide opportunities for candidates to explore the more advanced features of the GDC. An inappropriate choice of calculator window made it difficult for candidates to assess and appreciate the behaviour of $u_n$ and $v_n$.