Higher January 2023 Paper 1R Q24
24 An arithmetic sequence has first term 8 and common difference 11
The sequence has \(k\) terms, where \(k \gt 21\)
The sum of the last 20 terms of the sequence is 10 170
Find the value of \(k\)
Show clear algebraic working.
(5)
| Scheme | Marks |
|---|---|
| \((S_{20} =)\;10[2A + 19 \times 11] = 10\,170\) oe (where \(A\) is the \(u_{(k-19)}\) th term) | M1 |
| \(A = \left(\dfrac{10\,170}{10} - 19 \times 11\right) \div 2\;(= 404)\) | M1 |
| \(8 + (P - 1)11 = \text{``}{404}\text{''}\) oe (where \(P\) is the number of terms from 20 to the end) | M1 |
| \(P = \dfrac{\text{``}{404}\text{''} - 8 + 11}{11}\;(= 37)\) | M1 |
| Working required Answer: 56 | A1 |
| (5) | |
| (5 marks) |
Notes
A1: dep on M1
M2 for \(8 + 11 \times (k - 20) = \text{``}{404}\text{''}\) (in place of the 3rd and 4th M1)
| Scheme | Marks |
|---|---|
\((S_k =)\;\dfrac{k}{2}[2 \times 8 + (k - 1)11]\) or \((S_{k-20} =)\;\dfrac{(k - 20)}{2}[2 \times 8 + (k - 21)11]\) or \((u_{k-19} =)\;8 + 11(k - 20)\) or \((u_k =)\;8 + 11(k - 1)\) (allow use of letter other than k) | M1 |
\((S_k =)\;\dfrac{k}{2}[2 \times 8 + (k - 1)11]\) and \((S_{k-20} =)\;\dfrac{(k - 20)}{2}[2 \times 8 + (k - 21)11]\) or \((u_{k-19} =)\;8 + 11(k - 20)\) and \((u_k =)\;8 + 11(k - 1)\) | M1 |
\(10\,170 = \dfrac{k}{2}[\text{``}{16}\text{''} + (k - 1)11] - \dfrac{(k - 20)}{2}[\text{``}{16}\text{''} + (k - 21)11]\) oe or \(10\,170 = \dfrac{20}{2}\big([8 + 11(k - 20)] + [8 + 11(k - 1)]\big)\) oe | M1 |
eg \(10\,170 = 160 + \dfrac{11}{2}[40k - 420]\) oe eg \(440k = 24\,640\) or \(2240 = 40k\) oe | M1 |
| Working required Answer: 56 | A1 |
Notes
M1: for \(S_k\) or \(S_{k-20}\) or \(u_k\) or \(u_{k-19}\)
a and d must be substituted correctly
M1: For correct expressions for both \(S_k\) and \(S_{k-20}\) or \(u_k\) and \(u_{k-19}\)
M1: Expanding to obtain a linear equation and collecting terms in \(k\)
A1: dep on M1