Hint
Subtract the two term equations. For the final part, test the integer immediately before your proposed answer.
Worked solution
(a)
\(a+3d=14\) and \(a+8d=34\). Subtracting gives \(\displaystyle 5d=20\), so \(\displaystyle d=4\) and \(\displaystyle a=14-12=2\).
Answer: \(a=2,\ d=4\).
(b)
\(u_{20}=2+19(4)=78\). Hence \(\displaystyle S_{20}= \frac{20(2+ 78)}{2}= 800\).
Answer: 800.
(c)
\(u_n=2+4(n-1)=4n-2\). Thus 4n−2>100 means n>25.5. The least integer is \(\displaystyle 26: u_{25}=98\le 100\) and \(\displaystyle u_{26}=102>100\). Since d>0, no earlier index succeeds.
Answer: \(n=26\).
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) · 3 marks
- 1 mark: Write the two correctly indexed term equations.
- 1 mark: Subtract to obtain \(\displaystyle d=4\).
- 1 mark: Substitute to obtain \(\displaystyle a=2\).
Alternative methods: Elimination using any equivalent pair of valid equations is accepted.
Common error: Using four differences to reach the fourth term.
(b) · 2 marks
- 1 mark: Use a correct AP sum formula with 20 terms.
- 1 mark: Evaluate the total as 800.
Alternative methods: Pairing the 20 terms into ten pairs of sum 80 is accepted.
Common error: Giving the twentieth term instead of the sum.
(c) · 3 marks
- 1 mark: Obtain the correct strict inequality in n.
- 1 mark: Select the integer 26.
- 1 mark: Verify the threshold and predecessor, or give an equivalent complete integer-inequality argument.
Alternative methods: A correctly justified comparison of consecutive terms is accepted.
Common error: Rounding 25.5 down or treating it as an allowed index.
Total: 8 marks.