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S5 Maths: Arithmetic and Geometric Sequences (Challenge, 10 Questions)
S5 Maths: Arithmetic and Geometric Sequences (Challenge, 10 Questions)
Ten challenge items mixing both types: sums to infinity, recurring decimals as fractions, the largest partial sum, alternating signs with a negative ratio, inserted means, comparing two sequences, triangle sides and compound interest.
Quick answer Answers in order: 16, 4/9, 780, n = 34 giving 1717, -96 and -63, a = 3 and term 8 = 384, 6 and 18, sum 65 difference 43, 10 12 14, 11576.25. Decide first whether the pattern adds or multiplies.
01 Paper details (check answers afterwards)
| Item | Detail |
|---|---|
| Questions | 10 questions |
| Suggested time | about 35 minutes |
| Difficulty mix | Challenge 10 |
| Chapter | Chapter: S5 Maths: Arithmetic and Geometric Sequences |
| Level | S5 |
02 Questions (10 items)
- ChallengeA geometric sequence has first term 8 and common ratio 1/2. Find the sum to infinity.
- ChallengeWrite the recurring decimal 0.444… as a fraction in lowest terms.
- ChallengeIn an arithmetic sequence the 5th term is 17 and the 12th term is 45. Find the sum of the first 20 terms.
- ChallengeFor the arithmetic sequence 100, 97, 94, … which n makes the sum of the first n terms largest, and what is that sum?
- ChallengeA geometric sequence has first term 3 and common ratio -2. Find the 6th term and the sum of the first 6 terms.
- ChallengeIn a geometric sequence the 3rd term is 12 and the 5th term is 48 with a positive ratio. Find the first term and the 8th term.
- ChallengeInsert two numbers between 2 and 54 so that the four numbers form a geometric sequence.
- ChallengeAn arithmetic sequence has a = 2 and d = 3; a geometric sequence has a = 2 and r = 3. Compare their 4th terms.
- ChallengeThe three sides of a triangle form an arithmetic sequence with common difference 2 and the perimeter is 36. Find the sides.
- ChallengeA principal of 10000 dollars earns 5% per year compound interest for 3 years. Find the amount.
03 Answers and one-line notes
| # | Answer | One-line note |
|---|---|---|
| 1 | 16 | When the ratio is between -1 and 1 the infinite sum is a/(1 – r), so 8 / 0.5 = 16. |
| 2 | 4/9 | Treat it as 0.4 + 0.04 + 0.004 + … with a = 0.4 and r = 0.1, so the sum is 0.4 / 0.9 = 4/9. |
| 3 | 780 | The gap of 28 covers 7 differences, so d = 4 and a = 1; then S20 = 10 x (2 + 76) = 780. |
| 4 | n = 34, largest sum 1717 | Only positive terms add to the total: the 34th term is 1 and the 35th is -2, so stop at 34 and the sum is 34 x 101/2 = 1717. |
| 5 | 6th term -96, sum -63 | The 6th term is 3 x (-2)^5 = -96 and the sum is 3 x (64 – 1)/(-3) = -63; a negative ratio alternates signs. |
| 6 | first term 3, 8th term 384 | 48 / 12 = 4 is the ratio squared, so r = 2; the first term is 12 / 4 = 3 and the 8th term is 3 x 2^7 = 384. |
| 7 | 6 and 18 | With ratio r, 2 x r^3 = 54 so r = 3, giving the middle terms 6 and 18. |
| 8 | sum 65, difference 43 | The arithmetic 4th term is 2 + 9 = 11 and the geometric 4th term is 2 x 27 = 54, so the sum is 65 and the difference 43. |
| 9 | 10, 12, 14 | Write the sides as m – 2, m and m + 2; 3m = 36 gives m = 12, so the sides are 10, 12 and 14. |
| 10 | 11576.25 dollars | Compound interest multiplies by 1.05 each year: 10000 x 1.05^3 = 10000 x 1.157625 = 11576.25 dollars. |
04 What to do after marking
- After marking, redo only the wrong items and write down which step went wrong (formula, unit, or misread number).
- Repeat the same item two days later; move to the next paper only when both attempts are correct.
- Parents only need the answer and note columns; there is no need to explain every item.
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