Consider the expression tn = 3n − 7.
- (i) Find its first, second, third, 12th, 18th and 50th terms.
- (ii) Which term of the sequence is 332?
- (iii) Is 557 a term of this sequence? Why or why not?
Step-by-step solution
Idea: Substitute n to find a term. To find the position of a number, solve 3n − 7 = that number; it is a term only if n is a natural number.
(i) Find its first, second, third, 12th, 18th and 50th terms.
- t1 = 3 × 1 − 7 = −4; t2 = 3 × 2 − 7 = −1; t3 = 3 × 3 − 7 = 2.1 mark
- t12 = 36 − 7 = 29; t18 = 54 − 7 = 47; t50 = 150 − 7 = 143.1 mark
(ii) Which term of the sequence is 332?
- 3n − 7 = 332 ⇒ 3n = 339 ⇒ n = 113.½ mark
- So 332 is the 113th term.½ mark
(iii) Is 557 a term of this sequence? Why or why not?
- 3n − 7 = 557 ⇒ 3n = 564 ⇒ n = 188.½ mark
- 188 is a natural number, so yes, 557 is a term: the 188th term.½ mark
Check: 3 × 113 − 7 = 339 − 7 = 332 ✓; 3 × 188 − 7 = 564 − 7 = 557 ✓.
Answer to write in the exam
(i)
tn = 3n − 7
t1 = −4, t2 = −1, t3 = 2
t12 = 36 − 7 = 29, t18 = 54 − 7 = 47, t50 = 150 − 7 = 143
(ii)
3n − 7 = 332 ⇒ 3n = 339 ⇒ n = 113
∴ 332 is the 113th term.
(iii)
3n − 7 = 557 ⇒ 3n = 564 ⇒ n = 188 (a natural number)
∴ Yes, 557 is the 188th term.
Common mistakes that cost marks
- Getting the sign wrong in the first terms: 3 − 7 = −4, not 4.
- Solving 3n − 7 = 332 as 3n = 325. Add 7 to both sides: 3n = 339.
- Assuming 557 cannot be a term because it is large or odd. Always solve for n and check whether it is a natural number.
How this can come in the exam
Which term of the sequence tn = 3n − 7 is 101?
- 34th
- 35th
- 36th
- 37th
Show answer
(C) 36th
3n − 7 = 101 ⇒ 3n = 108 ⇒ n = 36.
Is 200 a term of the sequence tn = 3n − 7? Why or why not?
Show answer
3n − 7 = 200 ⇒ 3n = 207 ⇒ n = 69 (1 mark). 69 is a natural number, so 200 is the 69th term (1 mark).Try one yourself
For tn = 4n − 9, find the 1st, 10th and 25th terms. Is 391 a term?
Show answer
−5, 31, 91. 4n − 9 = 391 ⇒ n = 100, so yes, 391 is the 100th term.
More questions like this
- Consider the sequence 1, 4, 7, 10, 13, … . The nth term is tn = 3n − 2 (verify this for yourself).
- 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.
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