A sequence is given by the recursive rule t1 = −5, tn+1 = tn + 3 for n ≥ 1. Find the first five terms of the sequence. Is 52 a term of this sequence? If so, which term is it?
Answer: First five terms: −5, −2, 1, 4, 7. Explicit form tn = 3n − 8; 3n − 8 = 52 gives n = 20, so 52 is the 20th term.
Step-by-step solution
Idea: Each term is 3 more than the one before, starting at −5. So the nth term is −5 plus (n − 1) threes; solve that equal to 52.
- t2 = t1 + 3 = −5 + 3 = −2; t3 = −2 + 3 = 1; t4 = 4; t5 = 7.1 mark
- To reach the nth term we add 3 a total of (n − 1) times: tn = −5 + 3(n − 1) = 3n − 8.1 mark
- 3n − 8 = 52 ⇒ 3n = 60 ⇒ n = 20, a natural number.½ mark
- So yes, 52 is a term: the 20th term.½ mark
First five terms: −5, −2, 1, 4, 7. Yes, 52 is the 20th term.
Check: 3 × 20 − 8 = 52 ✓. Counting: from −5 to 52 is 57 = 19 steps of 3, so 1 + 19 = 20th term ✓.
Answer to write in the exam
t2 = −5 + 3 = −2, t3 = 1, t4 = 4, t5 = 7
∴ First five terms: −5, −2, 1, 4, 7
tn = −5 + 3(n − 1) = 3n − 8
3n − 8 = 52 ⇒ 3n = 60 ⇒ n = 20
∴ Yes, 52 is the 20th term.
Common mistakes that cost marks
- Writing −5 + 3 = −8 (adding 3 in the wrong direction). −5 + 3 = −2.
- Using 3n instead of 3(n − 1): the first term already exists, so only n − 1 threes are added.
- Misreading tn+1 = tn + 3 as ‘the term is n + 1 + 3′. It means: the next term is the current term plus 3.
How this can come in the exam
MCQ (1 mark)
If t1 = −5 and tn+1 = tn + 3, then t10 is
- 22
- 25
- 19
- 30
Show answer
(A) 22
3 × 10 − 8 = 22.
Short answer (2 marks)
A sequence has t1 = 40 and tn+1 = tn − 6 for n ≥ 1. Is −20 a term? If so, which?
Show answer
tn = 40 − 6(n − 1) = 46 − 6n (1 mark). 46 − 6n = −20 ⇒ n = 11, so −20 is the 11th term (1 mark).Try one yourself
t1 = 7 and tn+1 = tn + 4 for n ≥ 1. Write the first five terms. Is 87 a term?
Show answer
7, 11, 15, 19, 23. tn = 4n + 3; 4n + 3 = 87 ⇒ n = 21. Yes, the 21st term.
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