Hint

The sequence uses coefficients, not the polynomial terms. An increasing progression requires a positive common difference.

Worked solution

(a)

The first three coefficients are 1, 6k and \(\displaystyle 15k^{2}\). Twice the middle AP term equals the sum of the outer terms, so \(\displaystyle 12k=1+15k^{2}\), giving the stated quadratic.

Answer: \(2(6k)=1+15k^2\), hence \(15k^2-12k+1=0\).

(b)

The quadratic formula gives \(\displaystyle k= \frac{12\pm \sqrt{144- 60}}{30}= \frac{12\pm 2\sqrt{21}}{30}= \frac{6\pm \sqrt{21}}{15}\). Both are real and both are positive, so positivity alone cannot select a branch.

Answer: \(\displaystyle k= \frac{6\pm \sqrt{21}}{15}\).

(c)

The difference is \(\displaystyle d= 6k- 1= \frac{7\pm 2\sqrt{21}}{5}\). Since \(\displaystyle 2\sqrt{21}>7\), the minus branch has negative difference and the plus branch positive difference. Increasing order therefore selects the plus value of k and \(\displaystyle d= \frac{7+ 2\sqrt{21}}{5}\).

Answer: \(\displaystyle k= \frac{6+ \sqrt{21}}{15}\), \(\displaystyle d= \frac{7+ 2\sqrt{21}}{5}\).

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: Extract the three correct coefficient values.
• 1 mark: Use the AP relation between them.
• 1 mark: Obtain \(\displaystyle 15k^{2}-12k+1=0\).

Alternative methods: Equating successive differences is accepted.

Common error: Including x and \(\displaystyle x^{2}\) in the progression data or using a GP product condition.

(b) · 2 marks

• 1 mark: Use the quadratic formula or complete the square correctly.
• 1 mark: Give both exact roots \(\displaystyle \frac{6\pm \sqrt{21}}{15}\).

Alternative methods: Equivalent exact surd expressions are accepted.

Common error: Rejecting a branch without using the remaining condition.

(c) · 3 marks

• 1 mark: Express the common difference using the coefficient values.
• 1 mark: Select the plus parameter branch using the sign of the difference.
• 1 mark: Give the exact difference \(\displaystyle \frac{7+ 2\sqrt{21}}{5}\) with the rejected branch correctly accounted for.

Alternative methods: Compare both pairs of consecutive coefficients directly.

Common error: Selecting the plus branch solely because k must be positive; both roots are positive.

Total: 8 marks.