| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Arithmetic Sequences and Series |
| Type | Find term or common difference |
| Difficulty | Moderate -0.3 Part (a) involves straightforward application of the arithmetic series sum formula S_n = n/2[2a + (n-1)d] to find the first term, then solving a linear inequality. Part (b) requires substituting into a quadratic expression and solving a quadratic equation, followed by algebraic verification. All techniques are standard C1 procedures with no novel insight required, making this slightly easier than average but not trivial due to the multi-step nature and algebraic manipulation in part (b). |
| Spec | 1.04e Sequences: nth term and recurrence relations1.04h Arithmetic sequences: nth term and sum formulae |
| Answer | Marks | Guidance |
|---|---|---|
| (a)(i) \(\frac{20}{2}[2a + (19 \times 7)] = 530\) | M1 | |
| \(2a + 133 = 53\), so \(a = -40\) | M1 A1 | |
| (a)(ii) \(= -40 + 7k = -40 + 42 = 2\) | M1 A1 | |
| (b)(i) \(u_1 = (1 + k)^2, u_2 = (2 + k)^2\) | B1 | |
| \((2 + k)^2 = 2(1 + k)^2\) | M1 | |
| \(4 + 4k + k^2 = 2 + 4k + 2k^2\) | ||
| \(k^2 = 2\) | M1 | |
| \(k > 0\), so \(k = \sqrt{2}\) | A1 | |
| (b)(ii) \(u_3 = (3 + \sqrt{2})^2 = 9 + 6\sqrt{2} + 2 = 11 + 6\sqrt{2}\) | M1 A1 | (11) |
**(a)(i)** $\frac{20}{2}[2a + (19 \times 7)] = 530$ | M1 |
$2a + 133 = 53$, so $a = -40$ | M1 A1 |
**(a)(ii)** $= -40 + 7k = -40 + 42 = 2$ | M1 A1 |
**(b)(i)** $u_1 = (1 + k)^2, u_2 = (2 + k)^2$ | B1 |
$(2 + k)^2 = 2(1 + k)^2$ | M1 |
$4 + 4k + k^2 = 2 + 4k + 2k^2$ | |
$k^2 = 2$ | M1 |
$k > 0$, so $k = \sqrt{2}$ | A1 |
**(b)(ii)** $u_3 = (3 + \sqrt{2})^2 = 9 + 6\sqrt{2} + 2 = 11 + 6\sqrt{2}$ | M1 A1 | (11)7. (a) An arithmetic series has a common difference of 7 .
Given that the sum of the first 20 terms of the series is 530 , find
\begin{enumerate}[label=(\roman*)]
\item the first term of the series,
\item the smallest positive term of the series.\\
(b) The terms of a sequence are given by
$$u _ { n } = ( n + k ) ^ { 2 } , \quad n \geq 1 ,$$
where $k$ is a positive constant.\\
Given that $u _ { 2 } = 2 u _ { 1 }$,
\item find the value of $k$,
\item show that $u _ { 3 } = 11 + 6 \sqrt { 2 }$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 Q7 [11]}}