June 2025 Paper 1 Q3
3
(a) Evaluate \[\sum_{r=1}^{4} \frac{1}{r}\] giving your answer as a fraction in its lowest terms. [1]
(b) Write the sum \(1+3+5+7+9\) in a similar way to the series in part (a). [2]
(c) Explain why the sum to infinity of \(1+3+5+7+9+\ldots\) is not well defined. [1]
| Scheme | Marks | AO |
|---|---|---|
| \(\dfrac{1}{1}+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}=\dfrac{25}{12}\) | B1 | 1.1 |
| [1] |
Notes
B1: Must be in lowest terms. Allow \(2\frac{1}{12}\) BC
| Scheme | Marks | AO |
|---|---|---|
| Sum \(= \sum_{r=1}^{5}(2r-1)\) or \(\sum_{r=0}^{4}(2r+1)\) | M1 A1 | 3.1a 2.5 |
| [2] |
Notes
M1: Correct function to generate odd numbers. Condone no brackets
A1: Correct limits for their expression using sigma notation
Brackets must be seen.
Do not allow \(\sum_{r=1}^{5}(r+(r-1))\), \(\sum_{r=1}^{5}(2n-1)\) or similar
| Scheme | Marks | AO |
|---|---|---|
| The series is increasing, so the sum of \(n\) terms forms a divergent sequence so there is not well defined sum to infinity | B1 | 2.4 |
| [1] |
Notes
B1: Explains in technical language or otherwise that the sum gets bigger the more terms added in so the sum to infinity is not a number. See appendix
Appendix: exemplar responses for Q3(c)
| Response | Mark |
|---|---|
| As it is a divergent sequence [and ever increases] | B1 |
| The sequence is not convergent | B1 |
| The \(n\)th term value is constantly increasing | B1 |
| It is constantly increasing | B0 |
| It could go on indefinitely | B0 |
| Because we can’t sum to infinity arithmetic sequences | B0 |
| It is an arithmetic sequence | B0 |
| The sequence is not geometric sequence so the sum to infinity is infinity | B0 |