## Question 7:

### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $600 = 200 + (N-1)20 \Rightarrow N = ...$ | M1 | Use of 600 with a **correct** formula in an attempt to find $N$. A correct formula could be implied by a correct answer. |
| $N = 21$ | A1 | cso |

**Special cases:** $600 = 200 + 20N \Rightarrow N = 20$ is M0A0 (wrong formula); $\frac{600-200}{20} = 20 \therefore N = 21$ is M1A1 (correct formula implied)

**Listing:** All terms must be listed up to 600 and 21 correctly identified. Scores 2 if fully correct, 0 otherwise. **(2 marks)**

### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $S = \frac{21}{2}(2\times200 + 20\times20)$ or $\frac{21}{2}(200+600)$ | M1 | Use of correct sum formula with integer $n = N$ or $N-1$ from part (a) where $3 \leq N \leq 52$, and $a = 200$, $d = 20$ |
| $S = \frac{20}{2}(2\times200 + 19\times20)$ or $\frac{20}{2}(200+580)$ | | |
| $(= 8400$ or $7800)$ | A1 | Any correct un-simplified numerical expression with $n = 20$ or $n = 21$ (no follow through) |
| $600 \times (52 - ``N'')(= 18600)$ | M1 | $600 \times k$ where $k$ is an integer and $3 < k < 52$ |
| | A1ft | A correct un-simplified follow through expression with their $k$ consistent with $n$ so that $n + k = 52$ |
| Total is $27000$ | A1 | cao |

**Note:** For constant terms, may correctly use an AP sum with $d = 0$. **There are no marks in (b) for just finding $S_{52}$.** **(5 marks)**

**Recovery notes:** If $N = 20$ obtained in (a), then $S = \frac{20}{2}(2\times200+19\times20)+32\times600 = 7800+19200 = 27000$ — allow full marks in (b). If $N = 22$ obtained in (a), then $S = \frac{21}{2}(2\times200+20\times20)+31\times600 = 8400+18600 = 27000$ — allow full marks in (b). **[7 marks total]**

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