Which term of the AP : 21, 18, 15, … is −81? Also, is 0 a term of this AP? Give reasons for your answer.
Answer: −81 is the 35th term. Yes, 0 is a term: 21 − 3(n − 1) = 0 gives n = 8, a natural number, so 0 is the 8th term.
Step-by-step solution
Idea: a = 21, d = −3. Set tn = 21 + (n − 1)(−3) equal to each number and solve; it is a term only if n is a natural number.
- a = 21, d = 18 − 21 = −3, so tn = 21 − 3(n − 1) = 24 − 3n.½ mark
- 24 − 3n = −81 ⇒ 3n = 105 ⇒ n = 35. So −81 is the 35th term.1 mark
- 24 − 3n = 0 ⇒ 3n = 24 ⇒ n = 8.1 mark
- 8 is a natural number, so 0 is a term (the 8th): 21, 18, 15, 12, 9, 6, 3, 0.½ mark
−81 is the 35th term. Yes, 0 is the 8th term, because n = 8 is a natural number.
Check: 24 − 3 × 35 = 24 − 105 = −81 ✓; 24 − 3 × 8 = 0 ✓.
Answer to write in the exam
a = 21, d = 18 − 21 = −3
tn = 21 + (n − 1)(−3) = 24 − 3n
24 − 3n = −81 ⇒ 3n = 105 ⇒ n = 35
∴ −81 is the 35th term.
24 − 3n = 0 ⇒ 3n = 24 ⇒ n = 8 (a natural number)
∴ Yes, 0 is a term (the 8th term).
Common mistakes that cost marks
- Taking d = 3 instead of −3. The terms decrease, so the common difference is negative.
- Solving 21 − 3(n − 1) = −81 as 3(n − 1) = 60. Move carefully: 3(n − 1) = 21 + 81 = 102, n − 1 = 34, n = 35.
- Saying 0 cannot be a term because the terms are ‘positive’. The AP continues past 3 to 0 and below.
How this can come in the exam
MCQ (1 mark)
Which term of the AP 21, 18, 15, … is −30?
- 17th
- 18th
- 19th
- 16th
Show answer
(B) 18th
24 − 3n = −30 ⇒ n = 18.
Short answer (2 marks)
Is −50 a term of the AP 21, 18, 15, …? Give a reason.
Show answer
24 − 3n = −50 ⇒ 3n = 74 ⇒ n = 24.67 (1 mark). Not a natural number, so −50 is not a term (1 mark).Try one yourself
Which term of the AP 50, 46, 42, … is −14? Is 1 a term?
Show answer
tn = 54 − 4n. 54 − 4n = −14 ⇒ n = 17: the 17th term. 54 − 4n = 1 ⇒ n = 13.25, so 1 is not a term.
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