| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Arithmetic Sequences and Series |
| Type | Prove sum formula |
| Difficulty | Moderate -0.8 This is a straightforward C1 question testing standard arithmetic series knowledge. Part (a) is a bookwork proof that should be memorized. Parts (b)-(e) involve routine substitution into formulas with simple algebraic manipulation. The quadratic in part (d) factors easily, and part (e) requires only basic contextual reasoning. No novel problem-solving or insight required—purely procedural application of a standard topic. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.04h Arithmetic sequences: nth term and sum formulae1.04i Geometric sequences: nth term and finite series sum |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(2x+3y=26 \Rightarrow 3y=26-2x\), attempt to find \(m\) from \(y=mx+c\) | M1 | Complete method for finding gradient; may be implied by later correct work |
| Gradient \(= -\frac{2}{3}\) | A1 | Condone \(-\frac{2}{3}x\); ignore errors in constant term |
| Gradient of perpendicular \(= \frac{-1}{\text{their gradient}} = \frac{3}{2}\) | M1 | Uses \(m_1 \times m_2 = -1\) |
| Line through \((0,0)\): \(y = \frac{3}{2}x\) | A1 | Allow \(2y=3x\), \(y-3x/2=0\), \(2y-3x=0\) etc. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Solve \(y=\frac{3}{2}x\) with \(2x+3y=26\) to form equation in \(x\) or \(y\) | M1 | Eliminates variable between the two equations |
| Solve to obtain \(x=\) or \(y=\) | dM1 | Depends on previous M; attempts to solve |
| \(x=4\) or \(y=6\) | A1 | |
| \(B = \left(0, \frac{26}{3}\right)\) used or stated | B1 | \(y\)-coordinate of \(B\) is \(\frac{26}{3}\); must be used or stated in (b) |
| Area \(= \frac{1}{2} \times 4 \times \frac{26}{3}\) (Method 1: \(\frac{1}{2}\times OB \times x\)-coordinate of \(C\)) | dM1 | Depends on previous M; complete method to find area of triangle \(OBC\) |
| \(= \frac{52}{3}\) | A1 | cao with integer numerator and denominator; also accept \(\frac{104}{6}\) or \(\frac{1352}{78}\) |
# Question 9:
## Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $2x+3y=26 \Rightarrow 3y=26-2x$, attempt to find $m$ from $y=mx+c$ | M1 | Complete method for finding gradient; may be implied by later correct work |
| Gradient $= -\frac{2}{3}$ | A1 | Condone $-\frac{2}{3}x$; ignore errors in constant term |
| Gradient of perpendicular $= \frac{-1}{\text{their gradient}} = \frac{3}{2}$ | M1 | Uses $m_1 \times m_2 = -1$ |
| Line through $(0,0)$: $y = \frac{3}{2}x$ | A1 | Allow $2y=3x$, $y-3x/2=0$, $2y-3x=0$ etc. |
## Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Solve $y=\frac{3}{2}x$ with $2x+3y=26$ to form equation in $x$ or $y$ | M1 | Eliminates variable between the two equations |
| Solve to obtain $x=$ or $y=$ | dM1 | Depends on previous M; attempts to solve |
| $x=4$ or $y=6$ | A1 | |
| $B = \left(0, \frac{26}{3}\right)$ used or stated | B1 | $y$-coordinate of $B$ is $\frac{26}{3}$; must be used or stated in (b) |
| Area $= \frac{1}{2} \times 4 \times \frac{26}{3}$ (Method 1: $\frac{1}{2}\times OB \times x$-coordinate of $C$) | dM1 | Depends on previous M; complete method to find area of triangle $OBC$ |
| $= \frac{52}{3}$ | A1 | cao with integer numerator and denominator; also accept $\frac{104}{6}$ or $\frac{1352}{78}$ |9. An arithmetic series has first term $a$ and common difference $d$.
\begin{enumerate}[label=(\alph*)]
\item 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 $n > 21$.
\item Find the amount Sean repays in the 21st month.
Over the $n$ months, he repays a total of $\pounds 5000$.
\item Form an equation in $n$, and show that your equation may be written as
$$n ^ { 2 } - 150 n + 5000 = 0$$
\item Solve the equation in part (c).
\item State, with a reason, which of the solutions to the equation in part (c) is not a sensible solution to the repayment problem.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 Q9}}