Why is the nth term of an arithmetic sequence a + (n - 1)d?

The first term is already term 1, so moving from term 1 to term n takes only n - 1 steps of the common difference; using a + nd jumps one step too far and shifts every answer.

When does an infinite geometric series have a sum?

Only when the common ratio r satisfies |r| below 1, so the terms approach 0 and the sum converges to a / (1 - r); if |r| >= 1 the terms do not tend to 0 and no finite sum exists.

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

2026-11-08

Tutor Prep Guide · S5 Mathematics

This error clinic collects the ten commonest slips in S5 sequences: the wrong nth-term formula, a missing division by 2 in sums, signs with a negative ratio, term counts without the +1, recovering the common difference from two terms, mixing up geometric and arithmetic means, and the convergence condition for infinite series, each with the typical wrong answer, the correct one and a note.

Quick answer Answers in order: 58, 525, 162, 16, 72, x = 9, 16, 250, 80, -11. Write down your original wrong step and the formula it broke before redoing the question.

01 Paper details (check answers afterwards)

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

02 Questions (10 items)

Error redoRetry: an arithmetic sequence has first term 4 and common difference 6. Find the 10th term. (A student answered 64.)
Error redoRetry: first term 7, common difference 4, find the sum of the first 15 terms. (A student answered 555.)
Error redoRetry: the geometric sequence is 2, -6, 18, … Find the 5th term. (A student answered -162.)
Error redoRetry: how many terms are there in the arithmetic sequence 5, 9, 13, …, 65? (A student answered 15.)
Error redoRetry: the 4th term is 17 and the 9th term is 42. Find the 15th term. (A student answered 77.)
Error redoRetry: 3, x, 27 form a geometric sequence with x > 0. Find x. (A student answered 15.)
Error redoRetry: find the sum of the infinite geometric series 8 + 4 + 2 + … (A student answered “infinity”.)
Error redoRetry: first term 7, last term 43, 10 terms in total. Find the sum. (A student answered 500.)
Error redoRetry: first term 2, common ratio 3, find the sum of the first 4 terms. (A student answered 81.)
Error redoRetry: first term 10, common difference -3. Find the 8th term. (A student answered -14.)

03 Answers and one-line notes

#	Answer	One-line note
1	58	The nth term is a + (n – 1)d, so 4 + 9 x 6 = 58; 64 comes from 4 + 10 x 6, one step too many.
2	525	The last term is 7 + 14 x 4 = 63, so S = 15 x (7 + 63) / 2 = 525; 555 uses a wrong last term of 67.
3	162	The common ratio is -3, so the 5th term is 2 x (-3)^4 = 162; an even power makes the sign positive.
4	16	The common difference is 4, so the number of terms is (65 – 5) / 4 + 1 = 16; forgetting the +1 drops the last term.
5	72	Five steps apart gives d = (42 – 17) / 5 = 5 and a = 17 – 3 x 5 = 2, so the 15th term is 2 + 14 x 5 = 72; 77 uses 2 + 15 x 5.
6	x = 9	The geometric mean satisfies x^2 = 3 x 27 = 81, so x = 9; 15 is the arithmetic mean, which belongs to a different sequence.
7	16	The ratio is 1/2 and \|r\| below 1 so the series converges: S = 8 / (1 – 1/2) = 16; the terms shrink towards 0.
8	250	S = n(a + l) / 2 = 10 x (7 + 43) / 2 = 250; 500 forgets the final division by 2.
9	80	S4 = a(r^4 – 1) / (r – 1) = 2 x (81 – 1) / 2 = 80; 81 forgets to subtract the first power in the numerator.
10	-11	The 8th term is 10 + 7 x (-3) = -11; -14 comes from 10 + 8 x (-3), one extra step of -3.

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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