Prove sum formula - Arithmetic Sequences and Series

Edexcel C12 2015 January Q5

7 marks Moderate -0.8

5. (a) Prove that the sum of the first $n$ terms of an arithmetic series is given by the formula $$S _ { n } = \frac { n } { 2 } [ 2 a + ( n - 1 ) d ]$$ where $a$ is the first term of the series and $d$ is the common difference between the terms.
(b) Find the sum of the integers which are divisible by 7 and lie between 1 and 500

Edexcel C1 2005 June Q9

13 marks Moderate -0.8

9. An arithmetic series has first term $a$ and common difference $d$.

Prove that the sum of the first $n$ terms of the series is $$\frac { 1 } { 2 } n [ 2 a + ( n - 1 ) d ] .$$ Sean repays a loan over a period of $n$ months. His monthly repayments form an arithmetic sequence. He repays $\pounds 149$ in the first month, $\pounds 147$ in the second month, $\pounds 145$ in the third month, and so on. He makes his final repayment in the $n$th month, where $n > 21$.
Find the amount Sean repays in the 21st month. Over the $n$ months, he repays a total of $\pounds 5000$.
Form an equation in $n$, and show that your equation may be written as $$n ^ { 2 } - 150 n + 5000 = 0 .$$
Solve the equation in part (c).
State, with a reason, which of the solutions to the equation in part (c) is not a sensible solution to the repayment problem.

Edexcel C1 Q9

11 marks Moderate -0.8

9. An arithmetic series has first term $a$ and common difference $d$.

Prove that the sum of the first $n$ terms of the series is $\frac { 1 } { 2 } n [ 2 a + ( n - 1 ) d ] .$ (4)
A polygon has 16 sides. The lengths of the sides of the polygon, starting with the shortest side, form an arithmetic sequence with common difference $d \mathrm {~cm}$.
The longest side of the polygon has length 6 cm and the perimeter of the polygon is 72 cm .
Find
the length of the shortest side of the polygon,
(5)
the value of $d$.
(2) $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ \end{tabular} & Leave blank
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Leave blank
\begin{center} \begin{tabular}{|l|l|} \hline \begin{tabular}{l}

Edexcel P2 2020 January Q8

7 marks Moderate -0.8

8. (i) An arithmetic series has first term $a$ and common difference $d$. Prove that the sum to $n$ terms of this series is $$\frac { n } { 2 } \{ 2 a + ( n - 1 ) d \}$$ (ii) A sequence $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$ is given by $$u _ { n } = 5 n + 3 ( - 1 ) ^ { n }$$ Find the value of

$u _ { 5 }$
$\sum _ { n = 1 } ^ { 59 } u _ { n }$

Edexcel P2 2018 Specimen Q5

11 marks Easy -1.2

An arithmetic series has first term $a$ and common difference $d$.

Prove that the sum of the first $n$ terms of the series is $$\frac { 1 } { 2 } n [ 2 a + ( n - 1 ) d ]$$ A company, which is making 200 mobile phones each week, plans to increase its production. The number of mobile phones produced is to be increased by 20 each week from 200 in week 1 to 220 in week 2, to 240 in week 3 and so on, until it is producing 600 in week $N$.
Find the value of $N$ The company then plans to continue to make 600 mobile phones each week.
Find the total number of mobile phones that will be made in the first 52 weeks starting from and including week 1.

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Edexcel Paper 1 2022 June Q13

7 marks Easy -1.2

1. In an arithmetic series, the first term is $a$ and the common difference is $d$.

Show that $$S _ { n } = \frac { n } { 2 } [ 2 a + ( n - 1 ) d ]$$ (ii) James saves money over a number of weeks to buy a printer that costs $\pounds 64$ He saves $\pounds 10$ in week $1 , \pounds 9.20$ in week $2 , \pounds 8.40$ in week 3 and so on, so that the weekly amounts he saves form an arithmetic sequence. Given that James takes $n$ weeks to save exactly $\pounds 64$

show that $$n ^ { 2 } - 26 n + 160 = 0$$
Solve the equation $$n ^ { 2 } - 26 n + 160 = 0$$
Hence state the number of weeks James takes to save enough money to buy the printer, giving a brief reason for your answer.

Edexcel C1 Q6

10 marks Moderate -0.8

6. (a) An arithmetic series has first term $a$ and common difference $d$. Prove that the sum of the first $n$ terms of the series is $\frac { 1 } { 2 } n [ 2 a + ( n - 1 ) d ]$. A company made a profit of $\pounds 54000$ in the year 2001. A model for future performance assumes that yearly profits will increase in an arithmetic sequence with common difference $\pounds d$. This model predicts total profits of $\pounds 619200$ for the 9 years 2001 to 2009 inclusive.
(b) Find the value of $d$. Using your value of $d$,
(c) find the predicted profit for the year 2011.

Edexcel C1 Q9

12 marks Moderate -0.8

9. (a) Prove that the sum of the first $n$ terms of an arithmetic series with first term $a$ and common difference $d$ is given by $$\frac { 1 } { 2 } n [ 2 a + ( n - 1 ) d ] .$$ A novelist begins writing a new book. She plans to write 16 pages during the first week, 18 during the second and so on, with the number of pages increasing by 2 each week. Find, according to her plan,
(b) how many pages she will write in the fifth week,
(c) the total number of pages she will write in the first five weeks.
(d) Using algebra, find how long it will take her to write the book if it has 250 pages.

Edexcel C2 Q8

13 marks Moderate -0.8

8. (a) An arithmetic series has first term $a$ and common difference $d$. Prove that the sum of the first $n$ terms of the series is $\frac { 1 } { 2 } n [ 2 a + ( n - 1 ) d ]$. A company made a profit of $\pounds 54000$ in the year 2001. A model for future performance assumes that yearly profits will increase in an arithmetic sequence with common difference $\pounds d$. This model predicts total profits of $\pounds 619200$ for the 9 years 2001 to 2009 inclusive.
(b) Find the value of $d$. Using your value of $d$,
(c) find the predicted profit for the year 2011. An alternative model assumes that the company's yearly profits will increase in a geometric sequence with common ratio 1.06 . Using this alternative model and again taking the profit in 2001 to be $\pounds 54000$,
(d) find the predicted profit for the year 2011.

Edexcel C1 Q5

6 marks Moderate -0.8

The $r$ th term of an arithmetic series is $( 2 r - 5 )$.
1. Write down the first three terms of this series.
2. State the value of the common difference.
3. Show that $\sum _ { r = 1 } ^ { n } ( 2 r - 5 ) = n ( n - 4 )$.
\begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{466833b9-730d-424c-b33b-dd93a14ab21d-02_326_618_294_429}
\end{figure} Figure 1 shows a sketch of the curve with equation $y = \mathrm { f } ( x )$. The curve crosses the $x$-axis at the points $( 2,0 )$ and $( 4,0 )$. The minimum point on the curve is $P ( 3 , - 2 )$. In separate diagrams sketch the curve with equation
$y = - \mathrm { f } ( x )$,
$y = \mathrm { f } ( 2 x )$. On each diagram, give the coordinates of the points at which the curve crosses the $x$-axis, and the coordinates of the image of $P$ under the given transformation.
7. The curve $C$ has equation $y = 4 x ^ { 2 } + \frac { 5 - x } { x } , x \neq 0$. The point $P$ on $C$ has $x$-coordinate 1 .
Show that the value of $\frac { \mathrm { d } y } { \mathrm {~d} x }$ at $P$ is 3 .
Find an equation of the tangent to $C$ at $P$. This tangent meets the $x$-axis at the point $( k , 0 )$.
Find the value of $k$.
8. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{466833b9-730d-424c-b33b-dd93a14ab21d-02_483_974_280_1644}
\end{figure} The points $A ( 1,7 ) , B ( 20,7 )$ and $C ( p , q )$ form the vertices of a triangle $A B C$, as shown in Figure 2. The point $D ( 8,2 )$ is the mid-point of $A C$.
Find the value of $p$ and the value of $q$. The line $l$, which passes through $D$ and is perpendicular to $A C$, intersects $A B$ at $E$.
Find an equation for $l$, in the form $a x + b y + c = 0$, where $a$, b and $c$ are integers.
Find the exact $x$-coordinate of $E$.
9. The gradient of the curve $C$ is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = ( 3 x - 1 ) ^ { 2 }$$ The point $P ( 1,4 )$ lies on $C$.
Find an equation of the normal to $C$ at $P$.
Find an equation for the curve $C$ in the form $y = \mathrm { f } ( x )$.
Using $\frac { \mathrm { d } y } { \mathrm {~d} x } = ( 3 x - 1 ) ^ { 2 }$, show that there is no point on $C$ at which the tangent is parallel to the line $y = 1 - 2 x$.
10. Given that $$\mathrm { f } ( x ) = x ^ { 2 } - 6 x + 18 , \quad x \geq 0 ,$$
express $\mathrm { f } ( x )$ in the form $( x - a ) ^ { 2 } + b$, where $a$ and $b$ are integers. The curve $C$ with equation $y = \mathrm { f } ( x ) , x \geq 0$, meets the $y$-axis at $P$ and has a minimum point at $Q$.
Sketch the graph of $C$, showing the coordinates of $P$ and $Q$. The line $y = 41$ meets $C$ at the point $R$.
Find the $x$-coordinate of $R$, giving your answer in the form $p + q \sqrt { } 2$, where $p$ and $q$ are integers. Materials required for examination
Mathematical Formulae (Green) Paper Reference(s)
6663/01 Core Mathematics C1
Advanced Subsidiary
Monday 23 May 2005 - Morning
Time: $\mathbf { 1 }$ hour $\mathbf { 3 0 }$ minutes Calculators may NOT be used in this examination. Write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Core Mathematics C1), the paper reference (6663), your surname, initials and signature. A booklet 'Mathematical Formulae and Statistical Tables' is provided. Full marks may be obtained for answers to ALL questions.
There are 10 questions in this question paper. The total mark for this paper is 75 .
Advice to Candidates
You must ensure that your answers to parts of questions are clearly labelled.
You must show sufficient working to make your methods clear to the Examiner. Answers without working may gain no credit. N23491A
1. (a) Write down the value of $8 ^ { \frac { 1 } { 3 } }$.
2. Find the value of $8 ^ { - \frac { 2 } { 3 } }$.
3. Given that $y = 6 x - \frac { 4 } { x ^ { 2 } } , x \neq 0$,
4. find $\frac { \mathrm { d } y } { \mathrm {~d} x }$,
5. find $\int y \mathrm {~d} x$.
$$x ^ { 2 } - 8 x - 29 \equiv ( x + a ) ^ { 2 } + b$$ where $a$ and $b$ are constants.
Find the value of $a$ and the value of $b$.
Hence, or otherwise, show that the roots of $$x ^ { 2 } - 8 x - 29 = 0$$ are $c \pm d \sqrt { } 5$, where $c$ and $d$ are integers to be found.
4. Figure 1 \includegraphics[max width=\textwidth, alt={}, center]{466833b9-730d-424c-b33b-dd93a14ab21d-04_552_796_328_1720} Figure 1 shows a sketch of the curve with equation $y = \mathrm { f } ( x )$. The curve passes through the origin $O$ and through the point $( 6,0 )$. The maximum point on the curve is $( 3,5 )$. On separate diagrams, sketch the curve with equation
$y = 3 \mathrm { f } ( x )$,
$y = \mathrm { f } ( x + 2 )$. On each diagram, show clearly the coordinates of the maximum point and of each point at which the curve crosses the $x$-axis.
5. Solve the simultaneous equations $$\begin{gathered} x - 2 y = 1 \\ x ^ { 2 } + y ^ { 2 } = 29 \end{gathered}$$

Edexcel C1 Q9

Moderate -0.8

9. An arithmetic series has first term $a$ and common difference $d$.

Prove that the sum of the first $n$ terms of the series is $$\frac { 1 } { 2 } n [ 2 a + ( n - 1 ) d ] .$$ Sean repays a loan over a period of $n$ months. His monthly repayments form an arithmetic sequence. He repays $\pounds 149$ in the first month, $\pounds 147$ in the second month, $\pounds 145$ in the third month, and so on. He makes his final repayment in the $n$th month, where $n > 21$.
Find the amount Sean repays in the 21st month. Over the $n$ months, he repays a total of $\pounds 5000$.
Form an equation in $n$, and show that your equation may be written as $$n ^ { 2 } - 150 n + 5000 = 0$$
Solve the equation in part (c).
State, with a reason, which of the solutions to the equation in part (c) is not a sensible solution to the repayment problem.

Edexcel C1 Q18

10 marks Moderate -0.8

(a) An arithmetic series has first term $a$ and common difference $d$. Prove that the sum of the first $n$ terms of the series is

$$\frac { 1 } { 2 } n [ 2 a + ( n - 1 ) d ] .$$ A company made a profit of $\pounds 54000$ in the year 2001. A model for future performance assumes that yearly profits will increase in an arithmetic sequence with common difference $\pounds d$. This model predicts total profits of $\pounds 619200$ for the 9 years 2001 to 2009 inclusive.
(b) Find the value of $d$. Using your value of $d$,
(c) find the predicted profit for the year 2011.

AQA Further Paper 2 2021 June Q4

7 marks Moderate -0.8

4

Show that $$( r + 1 ) ^ { 2 } - r ^ { 2 } = 2 r + 1$$ 4
Use the method of differences to show that $$\sum _ { r = 1 } ^ { n } ( 2 r + 1 ) = n ^ { 2 } + 2 n$$ 4
Verify that using the formula for $\sum _ { r = 1 } ^ { n } r$ gives the same result as that given in part (b).
[0pt] [3 marks]