Consider the sequence that is generated by the explicit formula sn = 5n − 2.
- (a) Can you write the first 6 terms of this sequence?
- (b) What is the 100th term? The 1000th term?
- (c) Let us check if the numbers 308 and 473 are terms of this sequence.
- (d) Can you explain why we need n to be a natural number?
Step-by-step solution
Idea: To find a term, substitute n. To test a number, set 5n − 2 equal to it and solve: the number is a term only if n comes out as a natural number.
(a) Can you write the first 6 terms of this sequence?
- s1 = 5 × 1 − 2 = 3, s2 = 5 × 2 − 2 = 8, s3 = 13.½ mark
- s4 = 18, s5 = 23, s6 = 28. (Each term is 5 more than the one before.)½ mark
(b) What is the 100th term? The 1000th term?
- s100 = 5 × 100 − 2 = 500 − 2 = 498.½ mark
- s1000 = 5 × 1000 − 2 = 5000 − 2 = 4998.½ mark
(c) Let us check if the numbers 308 and 473 are terms of this sequence.
- 308: 5n − 2 = 308 ⇒ 5n = 310 ⇒ n = 62. This is a natural number, so 308 is the 62nd term.½ mark
- The worked solution then tests 471: 5n − 2 = 471 ⇒ 5n = 473 ⇒ n = 94.6. This is not a natural number, so 471 is not a term.½ mark
- The number printed in the question is 473, so check it too: 5n − 2 = 473 ⇒ 5n = 475 ⇒ n = 95. So 473 is the 95th term. (The question and its working use different numbers; both are answered here.)½ mark
(d) Can you explain why we need n to be a natural number?
- n is a position: 1st, 2nd, 3rd, … There is no ‘94.6th’ term, and no 0th or negative position, so a number is a term only when n is 1, 2, 3, …½ mark
Check: 5 × 62 − 2 = 308 ✓; 5 × 95 − 2 = 473 ✓. Every term ends in 3 or 8, and 471 ends in 1, so it cannot be a term ✓.
Answer to write in the exam
(a)
sn = 5n − 2
s1 = 3, s2 = 8, s3 = 13, s4 = 18, s5 = 23, s6 = 28
∴ 3, 8, 13, 18, 23, 28
(b)
s100 = 5 × 100 − 2 = 498
s1000 = 5 × 1000 − 2 = 4998
(c)
5n − 2 = 308 ⇒ 5n = 310 ⇒ n = 62
∴ 308 is the 62nd term.
5n − 2 = 471 ⇒ 5n = 473 ⇒ n = 94.6 (not a natural number)
∴ 471 is not a term.
5n − 2 = 473 ⇒ 5n = 475 ⇒ n = 95
∴ 473 is the 95th term.
(d)
n is the position of a term: 1st, 2nd, 3rd, …
∴ n must be a natural number.
Common mistakes that cost marks
- Stopping at n = 94.6 and saying ‘471 is the 94.6th term’. A decimal answer means the number is not a term.
- Solving 5n − 2 = 308 as 5n = 306. Move −2 across as +2: 5n = 310.
- Giving the value of n (62) as ‘the term’. 62 is the position; 308 is the term.
How this can come in the exam
Which of these is a term of the sequence sn = 5n − 2?
- 250
- 253
- 255
- 257
Show answer
(B) 253
5n − 2 = 253 gives 5n = 255, n = 51, a natural number. The others give n = 50.4, 51.4, 51.8.
Is 600 a term of the sequence sn = 7n + 4? Give a reason.
Show answer
7n + 4 = 600 ⇒ 7n = 596 ⇒ n = 85.14… (1 mark). This is not a natural number, so 600 is not a term (1 mark).Try one yourself
For sn = 6n − 1, write the first four terms and decide whether 155 and 160 are terms.
Show answer
5, 11, 17, 23. 6n − 1 = 155 gives n = 26, so 155 is the 26th term. 6n − 1 = 160 gives n = 26.83…, so 160 is not a term.
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