How do you tell arithmetic from geometric?

An arithmetic sequence adds a fixed difference; a geometric sequence multiplies by a fixed ratio.

Why does the nth term formula use n - 1?

Going from the first term to the nth takes n - 1 steps, so the gap is applied n - 1 times.

What is the arithmetic sum formula?

The sum is n/2 times the first plus last term, that is n/2 times (2a + (n - 1)d).

Can a common ratio be negative?

Yes. A negative ratio makes the signs alternate, for example 2, -6, 18.

S5 Maths sequences paper 1: nth term and partial sums

2026-11-10

Tutor Prep Guide · S5 Maths Arithmetic and Geometric Sequences

Ten items start from the basic arithmetic and geometric formulas, then build partial sums and reverse questions.

Direct answers　Items 01 to 03 use the nth term formula, 04 to 06 partial sums and solving for n, 07 to 09 reverse problems on the difference or ratio, and 10 the mistake of treating a geometric sequence as arithmetic.

01　Paper details (check answers afterwards)

Item	Detail
Questions	10
Suggested time	25
Difficulty mix	Foundation 3 · Intermediate 3 · Challenge 3 · Error redo 1
Chapter	Chapter: S5 Maths Arithmetic and Geometric Sequences
Level	S5

02　Questions (10 items)

FoundationAn arithmetic sequence has first term 3 and common difference 4. Find the 10th term.
FoundationThe sequence 5, 8, 11, … is arithmetic. Find the 12th term.
FoundationA geometric sequence has first term 2 and common ratio 3. Find the 5th term.
IntermediateAn arithmetic sequence has first term 7 and common difference 3. Find the sum of the first 15 terms.
IntermediateFor the geometric sequence 3, 6, 12, … find the 8th term.
IntermediateAn arithmetic sequence has first term 12 and common difference -3. Which term first equals 0?
ChallengeAn arithmetic sequence has first term 7 and its first three terms add to 33. Find the 10th term.
ChallengeA geometric sequence has first term 5 and common ratio 2. Find the sum of the first 6 terms.
ChallengeIn an arithmetic sequence the 3rd term is 11 and the 7th term is 27. Find the 15th term.
Error redoError drill: a student treated 2, 6, 18, … as arithmetic and got 26 for the 5th term. What is the correct 5th term?

03　Answers and one-line notes

#	Answer	One-line note
1	39	The nth term is a + (n – 1)d, which gives 3 + 9 times 4, so 39.
2	38	The common difference is 3, so the 12th term is 5 + 11 times 3, which is 38.
3	162	The nth term is a times r to the power n – 1, so 2 times 3 to the fourth power, which is 162.
4	420	The sum is 15/2 times (14 + 42), which is 7.5 times 56, so 420.
5	384	The common ratio is 2, so the 8th term is 3 times 2 to the seventh power, which is 384.
6	The 5th term	Solving 12 + (n – 1)(-3) = 0 gives 3n = 15, so n = 5.
7	43	The three-term sum gives 21 + 3d = 33, so d = 4 and the 10th term is 7 + 36 = 43.
8	315	The sum is 5 times (2 to the sixth power minus 1) divided by 1, which is 5 times 63, so 315.
9	59	The four steps add 16, so the common difference is 4 and the first term is 3; the 15th term is 3 + 56 = 59.
10	162	The sequence has common ratio 3, so the 5th term is 2 times 3 to the fourth power, which is 162; 26 comes from wrongly adding 6 each time.

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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