Consider the sequence 1, 4, 7, 10, 13, … . The nth term is tn = 3n − 2 (verify this for yourself).
Answer: Putting n = 1, 2, 3, 4, 5 in 3n − 2 gives 1, 4, 7, 10, 13 ✓. Also 3n − 2 goes up by exactly 3 when n goes up by 1, just as the sequence does.
Step-by-step solution
Idea: To verify an explicit rule, substitute each position and compare with the given terms, and check that the rule grows in the same way as the sequence.
- t1 = 3 × 1 − 2 = 1, t2 = 3 × 2 − 2 = 4, t3 = 3 × 3 − 2 = 7, t4 = 10, t5 = 13. These match the given terms.1 mark
- Why it keeps working: moving from position n to n + 1 changes 3n − 2 into 3(n + 1) − 2 = 3n + 1, which is 3 more. The sequence also goes up by 3 each time, so the rule fits every term.1 mark
Verified: tₙ = 3n − 2 gives 1, 4, 7, 10, 13, … and increases by 3 each step, exactly like the sequence.
Answer to write in the exam
tn = 3n − 2
t1 = 1, t2 = 4, t3 = 7, t4 = 10, t5 = 13
tn+1 − tn = 3(n + 1) − 2 − (3n − 2) = 3
∴ tn = 3n − 2 gives 1, 4, 7, 10, 13, … (verified)
Common mistakes that cost marks
- Checking only the first term. 3n − 2 and n2 both give 1 at n = 1; check several terms.
- Writing the rule as 3n + 1 because the step is 3. The rule must give 1 (not 4) when n = 1.
- Calculating 3 × 4 − 2 as 3 × 2 = 6. Multiply first, then subtract.
How this can come in the exam
MCQ (1 mark)
The nth term of 5, 9, 13, 17, … is
- 4n + 5
- 4n + 1
- 5n − 1
- n + 4
Show answer
(B) 4n + 1
Step 4, and 4 × 1 + 1 = 5 ✓, 4 × 2 + 1 = 9 ✓.
Try one yourself
Verify that the nth term of 2, 7, 12, 17, … is 5n − 3.
Show answer
5 − 3 = 2, 10 − 3 = 7, 15 − 3 = 12, 20 − 3 = 17 ✓, and 5n − 3 grows by 5 each step, like the sequence.
More questions like this
- Find the first four terms of the sequence given by the recursive rule u1 = 1, un = 2un−1 + 3 for n ≥ 2. Is 133 a term of this sequence?
- Find the first four terms of the sequence given by the recursive rule s1 = 3, sn = sn−1 (sn−1 − 1) for n ≥ 2.
- A recursive rule or formula does not only have to involve the previous term — it could involve the previous two or more terms. The most famous example of such a sequence is V1, V2, V3, … where V1 = 1, V2 = 2, and Vn = Vn−1 + Vn−2 for n ≥ 3. So we get the sequence 1, 2, 3, 5, 8, 13, 21, 34, … where each term is obtained by adding the previous two. Can you write the next two terms of this sequence?
- Find the first five terms of the sequence in which the nth term is given by
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