Hint
The ratio is constant. Divide the second term by the first.
Worked solution
(a)
The ratio is the second term divided by the first; apply it n−1 times.
Answer: \(\displaystyle \frac{1}{4}\); \(3 \cdot 4^{1 - n}\)
(b)
Here 0<r<1, so \(\displaystyle r^{n}\) tends to zero; use the finite formula and its limit.
Answer: \(\displaystyle \frac{16383}{4096}\); \(4\)
(c)
Use the finite geometric-sum formula.
\(\displaystyle \frac{16383}{4096}\)
Answer: \(\displaystyle \frac{16383}{4096}\)
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit complete valid alternatives and consequential reasoning; an unexplained final answer does not earn a method mark.
(a) · 2 marks
- 1 mark: Use consecutive terms to find the ratio and form the general indexed term.
- 1 mark: Give all results requested in this part, with the correct signs, exact values, units, branch and boundary inclusion where applicable.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Do not use \(\displaystyle \frac{a}{1-r}\) for every geometric progression.
(b) · 2 marks
- 1 mark: Apply the finite geometric sum and use |r|<1 to justify its limit.
- 1 mark: Give all results requested in this part, with the correct signs, exact values, units, branch and boundary inclusion where applicable.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: The tail and the next term are different quantities.
(c) · 2 marks
- 1 mark: Use the finite geometric-sum formula.
- 1 mark: Give the correct requested result, retaining signs, units and any domain restrictions.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Do not use \(\displaystyle \frac{a}{1-r}\) for every geometric progression.
Total: 6 marks.