| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2007 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Arithmetic Sequences and Series |
| Type | Sequence defined by formula |
| Difficulty | Moderate -0.3 This is a standard C1 arithmetic sequence question requiring pattern recognition (4,7,10→3n+1), sum formula application, and solving a quadratic inequality. All techniques are routine for this level, though part (c) requires careful algebraic manipulation to reach the given form. Slightly easier than average due to scaffolded parts and standard methods throughout. |
| Spec | 1.04e Sequences: nth term and recurrence relations1.04h Arithmetic sequences: nth term and sum formulae |
| Answer | Marks | Guidance |
|---|---|---|
| (a) Recognising arithmetic series with first term 4 and common difference 3. (If not scored here, this mark may be given if seen elsewhere in the solution.) | B1 | |
| \(a + (n-1)d = 4 + 3(n-1)\) | \((= 3n + 1)\) | M1, A1 |
| (b) \(S_n = \frac{n}{2} [2a + (n-1)d] = \frac{10}{2} [8 + (10-1) \times 3] = 175\) | M1, A1, A1 | |
| (c) \(S_k < 1750\): \(\frac{k}{2} [8 + 3(k-1)] < 1750\) or \(S_{k+1} > 1750\): \(\frac{k+1}{2} [8 + 3k] > 1750\) | M1 | |
| \(3k^2 + 5k - 3500 < 0\) (or \(3k^2 + 11k - 3492 > 0\)) | M1, A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \((3k - 100)(k + 35) < 0\) | Requires use of correct inequality throughout (*) | A1cso |
| (d) \(\frac{100}{3}\) or equiv. seen (or \(\frac{97}{3}\)) | M1, A1 |
**(a)** Recognising arithmetic series with first term 4 and common difference 3. (If not scored here, this mark may be given if seen elsewhere in the solution.) | B1 |
$a + (n-1)d = 4 + 3(n-1)$ | $(= 3n + 1)$ | M1, A1 | (3 marks)
**(b)** $S_n = \frac{n}{2} [2a + (n-1)d] = \frac{10}{2} [8 + (10-1) \times 3] = 175$ | | M1, A1, A1 | (3 marks)
**(c)** $S_k < 1750$: $\frac{k}{2} [8 + 3(k-1)] < 1750$ or $S_{k+1} > 1750$: $\frac{k+1}{2} [8 + 3k] > 1750$ | | M1 |
$3k^2 + 5k - 3500 < 0$ (or $3k^2 + 11k - 3492 > 0$) | | M1, A1 |
(Allow equivalent 3-term versions such as $3k^2 + 5k = 3500$)
$(3k - 100)(k + 35) < 0$ | Requires use of correct inequality throughout (*) | A1cso | (4 marks)
**(d)** $\frac{100}{3}$ or equiv. seen (or $\frac{97}{3}$) | | M1, A1 | (2 marks)
**Total: 12 marks**
$k = 33$ (and no other values)
**Guidance:**
- (a) B1: Usually identified by $a = 4$ and $d = 3$. M1: Attempted use of term formula for arithmetic series, or… answer in the form $(3n + \text{constant})$, where the constant is a non-zero value. Answer for (a) does not require simplification, and a correct answer without working scores all 3 marks.
- (b) M1: Use of correct sum formula with $n = 9, 10$ or $11$. A1: Correct, perhaps unsimplified, numerical version. A1: 175. Alternative: (Listing and summing terms). M1: Summing 9, 10 or 11 terms. (At least 1st, 2nd and last terms must be seen). A1: Correct terms (perhaps implied by last term 31). A1: 175. Alternative: (Listing all sums). M1: Listing 9, 10 or 11 sums. (At least 4, 7, …, "last"). A1: Correct sums, correct finishing value 175. A1: 175. Alternative: (Using last term). M1: Using $S_n = \frac{n}{2}(a + l)$ with $T_9, T_{10}$ or $T_{11}$ as the last term. A1: Correct numerical version $\frac{10}{2}(4 + 31)$. A1: 175. Correct answer with no working scores 1 mark: 1,0,0.
- (c) For the first 3 marks, allow any inequality sign, or equals. 1st M: Use of correct sum formula to form inequality or equation in k, with the 1750. 2nd M: (Dependent on 1st M). Form 3-term quadratic in k. 1st A: Correct 3 terms. Allow credit for part (c) if valid work is seen in part (d).
- (d) Allow both marks for $k = 33$ seen without working. Working for part (d) must be seen in part (d), not part (c).
---9. Ann has some sticks that are all of the same length. She arranges them in squares and has made the following 3 rows of patterns:
Row 1 □
Row 2 □ 1
Row 3\\
\includegraphics[max width=\textwidth, alt={}, center]{fff086fd-f5d8-45b7-8db1-8b22ba5aab31-11_40_104_566_479}
She notices that 4 sticks are required to make the single square in the first row, 7 sticks to make 2 squares in the second row and in the third row she needs 10 sticks to make 3 squares.
\begin{enumerate}[label=(\alph*)]
\item Find an expression, in terms of $n$, for the number of sticks required to make a similar arrangement of $n$ squares in the $n$th row.
Ann continues to make squares following the same pattern. She makes 4 squares in the 4th row and so on until she has completed 10 rows.
\item Find the total number of sticks Ann uses in making these 10 rows.
Ann started with 1750 sticks. Given that Ann continues the pattern to complete $k$ rows but does not have sufficient sticks to complete the ( $k + 1$ )th row,
\item show that $k$ satisfies $( 3 k - 100 ) ( k + 35 ) < 0$.
\item Find the value of $k$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 2007 Q9 [12]}}