Hint

The ratio restriction selects a root of the coefficient equation. For the final part, compare tails at consecutive integer indices.

Worked solution

(a)

The coefficient is \(\displaystyle 10r^{2}= \frac{5}{2}\), so \(\displaystyle r^{2}= \frac{1}{4}\). Since \(\displaystyle 0< r< 1, r= \frac{1}{2}\). From \(\displaystyle \frac{a}{1- r}= 12\) we get \(\displaystyle a=6\).

Answer: \(\displaystyle r= \frac{1}{2}\), \(a=6\).

(b)

\(\displaystyle S_{4}= \frac{6(1- (\frac{1}{2})^{4})}{1- \frac{1}{2}}= 12(\frac{15}{16})= \frac{45}{4}\). Subtracting from the infinite sum gives \(\displaystyle 12- \frac{45}{4}= \frac{3}{4}\).

Answer: \(\displaystyle S_{4}= \frac{45}{4}\); tail \(\displaystyle \frac{3}{4}\).

(c)

The tail is \(\displaystyle R_{n}=12\cdot 2^{−n}\). Requiring \(\displaystyle R_{n}< \frac{1}{50}\) gives \(\displaystyle 2^{n}>600\). At \(\displaystyle n=9\) the tail is \(\displaystyle \frac{3}{128}\approx 0.0234375\), while at \(\displaystyle n=10\) it is \(\displaystyle \frac{3}{256}\approx 0.01171875\). The tails decrease strictly, so the least successful n is 10.

Answer: \(n=10\).

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) · 3 marks

• 1 mark: Use the correct binomial coefficient equation.
• 1 mark: Select \(\displaystyle r= \frac{1}{2}\) using the given ratio domain.
• 1 mark: Use the infinite sum to obtain \(\displaystyle a=6\).

Alternative methods: Equivalent elimination using the convergent GP formula is accepted.

Common error: Keeping the negative ratio root despite the stated positive domain.

(b) · 2 marks

• 1 mark: Give the exact finite sum \(\displaystyle \frac{45}{4}\).
• 1 mark: Give the distinct omitted tail \(\displaystyle \frac{3}{4}\).

Alternative methods: Adding \(\displaystyle 6+ 3+ \frac{3}{2}+ \frac{3}{4}\) directly is accepted.

Common error: Treating the fourth term as the whole omitted tail.

(c) · 3 marks

• 1 mark: Form the correct strict tail inequality.
• 1 mark: Obtain \(\displaystyle n=10\).
• 1 mark: Check the consecutive bounds and justify minimality.

Alternative methods: A logarithm calculation followed by a valid strict integer check is accepted.

Common error: Using an individual term threshold or rounding a real index down.

Total: 8 marks.