Why do sequences matter in S5?

S5 uses sequences for patterns, savings and compound interest, and population growth, and the DSE paper one regularly tests the nth term and the sum of the first n terms; telling arithmetic from geometric is the key step.

Why write the nth term formula first?

Once the nth term is written, any term, any number of terms and any total can be found by substitution; adding terms one by one is slow and fails when there are many.

S5 Maths: Arithmetic and Geometric Sequences (Intermediate, 10 Questions)

2026-11-08

Tutor Prep Guide · S5 Mathematics

Ten intermediate items: finding the first term and difference from two given terms, solving for n from a sum, setting three terms as m – d, m, m + d, and applying sequences to savings and growth.

Quick answer Answers in order: a = 3 d = 4, 735, 10, 405, r = 2 a = 3, 189, 7 12 17, term 32, 162 people, 1050 and 9300 dollars. For reverse items subtract the two given terms first.

01 Paper details (check answers afterwards)

Item	Detail
Questions	10 questions
Suggested time	about 30 minutes
Difficulty mix	Intermediate 10
Chapter	Chapter: S5 Maths: Arithmetic and Geometric Sequences
Level	S5

02 Questions (10 items)

IntermediateIn an arithmetic sequence the 3rd term is 11 and the 7th term is 27. Find the first term and the common difference.
IntermediateAn arithmetic sequence has first term 5, last term 65 and 21 terms. Find the total.
IntermediateFor the arithmetic sequence 3, 7, 11, … the sum of the first n terms is 210. Find n.
IntermediateA geometric sequence has first term 5 and common ratio 3. Find the 5th term.
IntermediateIn a geometric sequence the 2nd term is 6 and the 5th term is 48. Find the common ratio and the first term.
IntermediateA geometric sequence has first term 3 and common ratio 2. Find the sum of the first 6 terms.
IntermediateThree numbers form an arithmetic sequence with common difference 5 and add up to 36. Find them.
IntermediateA sequence has nth term a_n = 3n + 4. Is 100 a term of the sequence? If so, which term?
IntermediateTwo people know a piece of news on day 1, and each day the number is 3 times the day before. How many know it on day 5?
IntermediateMing saves 500 dollars in the first month and 50 dollars more each month. How much in month 12, and in total over 12 months?

03 Answers and one-line notes

#	Answer	One-line note
1	first term 3, common difference 4	The two terms are 4 differences apart: 27 – 11 = 16, so d = 4, and the first term is 11 – 2 x 4 = 3.
2	735	The sum is number of terms x (first + last)/2 = 21 x 70/2 = 735.
3	10	The sum is n/2 x [6 + 4(n – 1)] = n(2n + 1), and n(2n + 1) = 210 gives n = 10.
4	405	The 5th term is 5 x 3^4 = 5 x 81 = 405.
5	ratio 2, first term 3	48 / 6 = 8 is the ratio cubed, so r = 2, and the first term is 6 / 2 = 3.
6	189	The sum is 3 x (2^6 – 1)/(2 – 1) = 3 x 63 = 189.
7	7, 12, 17	Write them as m – 5, m and m + 5; the total is 3m = 36, so m = 12.
8	yes, the 32nd term	Solving 3n + 4 = 100 gives n = 32, a positive integer, so 100 is the 32nd term.
9	162 people	The numbers form a geometric sequence with a = 2 and r = 3, so day 5 is 2 x 3^4 = 162.
10	1050 dollars, 9300 dollars in total	The monthly amounts form an arithmetic sequence: month 12 is 500 + 11 x 50 = 1050, and the total is 12 x (500 + 1050)/2 = 9300.

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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