Hint
The next term changes \(\displaystyle S_{n}\) by exactly that term. A zero term can leave the partial sum unchanged.
Worked solution
(a)
\(u_n=18+(n-1)(-2)=20-2n\).
Answer: \(u_n=20-2n\).
(b)
20−2n>0 gives n<10. There are nine strictly positive terms, ending at \(\displaystyle u_{9}=2\). Their sum is \(\displaystyle \frac{9(18+ 2)}{2}= 90\); \(\displaystyle u_{10}=0\) is excluded.
Answer: Nine terms; sum 90.
(c)
\(\displaystyle S_{n}= n\{36- \frac{2(n- 1)\}}{2}= n(19- n)= 90.25- (n- 9.5)^{2}\). Among integers the nearest indices to 9.5 are 9 and 10; both give 90. Equivalently, \(\displaystyle S_{n}\) increases through the ninth term, is unchanged by the zero tenth term, and decreases thereafter.
Answer: 90 at \(n=9\) and \(n=10\).
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official examination-board mark scheme. Award stated criteria once each, up to the part maximum. Accept valid equivalent methods. Follow-through applies only where explicitly stated; do not award an accuracy criterion for an incorrect final result.
(a) · 2 marks
- 1 mark: Use first term plus (n−1) differences.
- 1 mark: Simplify to 20−2n.
Alternative methods: An equivalent expression 18−2(n−1) is accepted.
Common error: Writing 18−2n.
(b) · 3 marks
- 1 mark: Find exactly nine strictly positive terms.
- 1 mark: Use the correct finite-sum calculation.
- 1 mark: Give 90 and distinguish the zero tenth term from a positive term.
Alternative methods: Adding 18+16+…+2 directly is accepted.
Common error: Including zero as a strictly positive term.
(c) · 3 marks
- 1 mark: Obtain \(\displaystyle S_{n}=n(19-n)\), or compare signs of successive terms correctly.
- 1 mark: Give the greatest attainable sum 90.
- 1 mark: Identify both and only \(\displaystyle n=9\) and \(\displaystyle n=10\) with an integer or term-sign justification.
Alternative methods: The sign of S_(n+1)\(\displaystyle -S_{n}\)=u_(n+1) gives a complete alternative proof.
Common error: Reporting the continuous maximum 90.25 or only one of the two indices.
Total: 8 marks.