Explanation:
We are asked to find the
nth term rule for each linear sequence, and write it in the form
an + b, where *a* is the common difference and *b* is the constant (often found by plugging in n = 1).
A linear (arithmetic) sequence has a constant difference between consecutive terms. So:
1. Find the
common difference (*a*) = next term − previous term.
2. Then find *b* using the first term:
When n = 1, term = a(1) + b ⇒ b = first term − a.
Let’s go one by one:
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1. Sequence: –1, 0, 1, 2, 3, …
Common difference: 0 − (–1) = 1 → *a* = 1
First term (n=1) = –1
So: –1 = 1·1 + b → b = –1 − 1 = –2
→ nth term =
1n − 2 →
n − 2
✔ Check: n=1 → 1−2 = –1 ✔️; n=2 → 2−2 = 0 ✔️
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2. Sequence: 3, 5, 7, 9, 11, …
Difference: 5−3 = 2 → *a* = 2
First term = 3
3 = 2·1 + b → b = 3 − 2 = 1
→ nth term =
2n + 1
✔ Check: n=1 → 2+1=3 ✔️; n=3 → 6+1=7 ✔️
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3. Sequence: –2, 0, 2, 4, 6, …
Difference: 0 − (–2) = 2 → *a* = 2
First term = –2
–2 = 2·1 + b → b = –2 − 2 = –4
→ nth term =
2n − 4
✔ Check: n=1 → 2−4 = –2 ✔️; n=2 → 4−4 = 0 ✔️
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4. Sequence: 2, 5, 8, 11, 14, …
Difference: 5−2 = 3 → *a* = 3
First term = 2
2 = 3·1 + b → b = 2 − 3 = –1
→ nth term =
3n − 1
✔ Check: n=1 → 3−1=2 ✔️; n=4 → 12−1=11 ✔️
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5. Sequence: –2, 1, 4, 7, 10, …
Difference: 1 − (–2) = 3 → *a* = 3
First term = –2
–2 = 3·1 + b → b = –2 − 3 = –5
→ nth term =
3n − 5
✔ Check: n=1 → 3−5 = –2 ✔️; n=2 → 6−5 = 1 ✔️
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6. Sequence: 11, 12, 13, 14, 15, …
Difference: 12−11 = 1 → *a* = 1
First term = 11
11 = 1·1 + b → b = 10
→ nth term =
n + 10
✔ Check: n=1 → 1+10=11 ✔️; n=5 → 5+10=15 ✔️
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7. Sequence: 17, 18, 19, 20, 21, …
Difference = 1 → *a* = 1
First term = 17
17 = 1·1 + b → b = 16
→ nth term =
n + 16
✔ Check: n=1 → 1+16=17 ✔️
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8. Sequence: 13, 14, 15, 16, 17, …
Difference = 1 → *a* = 1
First term = 13
13 = 1·1 + b → b = 12
→ nth term =
n + 12
✔ Check: n=1 → 1+12=13 ✔️
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9. Sequence: –4, –3, –2, –1, 0, …
Difference = (–3) − (–4) = 1 → *a* = 1
First term = –4
–4 = 1·1 + b → b = –5
→ nth term =
n − 5
✔ Check: n=1 → 1−5 = –4 ✔️; n=5 → 5−5 = 0 ✔️
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10. Sequence: 17, 19, 21, 23, 25, …
Difference = 19−17 = 2 → *a* = 2
First term = 17
17 = 2·1 + b → b = 15
→ nth term =
2n + 15
✔ Check: n=1 → 2+15=17 ✔️; n=3 → 6+15=21 ✔️
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Now let’s list all answers in the required “an + b” form:
1. n − 2
2. 2n + 1
3. 2n − 4
4. 3n − 1
5. 3n − 5
6. n + 10
7. n + 16
8. n + 12
9. n − 5
10. 2n + 15
Final Answer:
1. n − 2
2. 2n + 1
3. 2n − 4
4. 3n − 1
5. 3n − 5
6. n + 10
7. n + 16
8. n + 12
9. n − 5
10. 2n + 15
Parent Tip: Review the logic above to help your child master the concept of sequences worksheet nth.
