Must you find the last term before summing?

The first-plus-last form needs it, but n/2 times (2a + (n - 1)d) works directly.

Can the sum formula give n?

Yes. Treat the sum as known, solve the quadratic and keep only the positive integer root.

How does a geometric mean differ from an arithmetic mean?

The arithmetic mean is the average of the two numbers; the geometric mean is the positive square root of their product.

Can the common ratio be a fraction?

Yes. A ratio of 1/2 halves each term, so the sequence decreases while staying positive.

S5 Maths sequences paper 2: sums and reverse problems

2026-11-10

Tutor Prep Guide · S5 Maths Arithmetic and Geometric Sequences

Ten items practise the sum formulas, recovering the first term and difference from a sum, geometric means and growth conditions.

Direct answers　Items 01 to 03 cover direct sums and the nth term, 04 to 06 an inequality test, a fractional ratio and a reverse sum, 07 to 09 a geometric mean, solving for n and a ratio, and 10 confusing a term with a sum.

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)

FoundationFor the arithmetic sequence 4, 9, 14, … find the sum of the first 10 terms.
FoundationFor the geometric sequence 1, 3, 9, … find the sum of the first 5 terms.
FoundationAn arithmetic sequence has nth term 5n – 2. Find the 20th term.
IntermediateFor 3, 7, 11, … how many terms are needed before the sum first exceeds 300?
IntermediateA geometric sequence has first term 8 and common ratio 1/2. Find the 6th term.
IntermediateAn arithmetic sequence has the sum of the first n terms equal to n squared plus 2n. Find the first term and the common difference.
ChallengeThree positive numbers form a geometric sequence with product 216 and middle term 6. Find the other two.
ChallengeAn arithmetic sequence has first term 6 and common difference 4; the sum of the first n terms is 240. Find n.
ChallengeA geometric sequence has first term 3 and its first four terms add to 45 with a positive integer ratio. Find the 5th term.
Error redoError drill: for an arithmetic sequence with first term 2 and common difference 3 a student gave 29 as the sum of the first 10 terms. What is correct?

03　Answers and one-line notes

#	Answer	One-line note
1	265	The sum is 5 times (8 + 45), which is 265.
2	121	The sum is (243 – 1) divided by 2, which is 121.
3	98	Substituting n = 20 gives 100 – 2 = 98; the first term is 3 and the common difference is 5.
4	13 terms	The sum equals n(2n + 1); n = 12 gives exactly 300 and n = 13 gives 351, so 13 terms are needed.
5	0.25	The 6th term is 8 times one half to the fifth power, which is 8 divided by 32, so 0.25.
6	First term 3, common difference 2	The first sum gives 3, and the second sum gives 8, so the second term is 5 and the difference is 2.
7	2 and 18	With middle term 6 and ratio 3 the three numbers are 2, 6 and 18, whose product is 216.
8	10	The sum is n times (2n + 4), and solving that equal to 240 gives n = 10.
9	48	Dividing by 3 requires 1 + r + r squared + r cubed to be 15, so r = 2 and the 5th term is 3 times 16, which is 48.
10	155	The value 29 is only the 10th term; the sum is 5 times (4 + 27), which is 155.

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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