| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Topic | Arithmetic Sequences and Series |
| Type | Prove sum formula |
| Difficulty | Moderate -0.8 This is a straightforward C1 arithmetic series question requiring standard formula derivation (bookwork proof), direct substitution into the nth term formula, forming and simplifying an equation using the sum formula, and solving a quadratic. All steps are routine applications of well-practiced techniques with no novel problem-solving required, making it easier than average but not trivial due to the multi-part structure and algebraic manipulation needed. |
| Spec | 1.01a Proof: structure of mathematical proof and logical steps1.04h Arithmetic sequences: nth term and sum formulae |
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
$$\frac{1}{2}n[2a + (n - 1)d].$$ [4]
\end{enumerate}
Sean repays a loan over a period of $n$ months. His monthly repayments form an arithmetic sequence.
He repays £149 in the first month, £147 in the second month, £145 in the third month, and so on. He makes his final repayment in the $n$th month, where $n > 21$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find the amount Sean repays in the 21st month. [2]
\end{enumerate}
Over the $n$ months, he repays a total of £5000.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Form an equation in $n$, and show that your equation may be written as
$$n^2 - 150n + 5000 = 0.$$ [3]
\item Solve the equation in part (c). [3]
\item State, with a reason, which of the solutions to the equation in part (c) is not a sensible solution to the repayment problem. [1]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 Q9 [13]}}