Math worksheet on finding the nth term of sequences with examples and a workout section.
A math worksheet from Corbett Maths titled "Sequences: nth term" featuring two questions with multiple sequences to find the nth term, a play button, a QR code, and links to videos on www.corbettmaths.com.
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Step-by-step solution for: Sequences: nth term Textbook Exercise - Corbettmaths
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Show Answer Key & Explanations
Step-by-step solution for: Sequences: nth term Textbook Exercise - Corbettmaths
Let's solve Question 1 and Question 2 step by step. We are asked to find the nth term of each arithmetic sequence.
---
For an arithmetic sequence, the nth term is given by:
$$
a_n = a + (n - 1)d
$$
Where:
- $ a $ = first term
- $ d $ = common difference (difference between consecutive terms)
- $ n $ = position in the sequence
We'll use this formula for all sequences where the pattern is linear (i.e., constant difference).
---
## ✔ Question 1: Find the nth term
- First term $ a = 5 $
- Common difference $ d = 8 - 5 = 3 $
- $ a_n = 5 + (n - 1) \cdot 3 = 5 + 3n - 3 = 3n + 2 $
✔ Answer: $ 3n + 2 $
---
- $ a = 9 $
- $ d = 14 - 9 = 5 $
- $ a_n = 9 + (n - 1) \cdot 5 = 9 + 5n - 5 = 5n + 4 $
✔ Answer: $ 5n + 4 $
---
- $ a = 1 $
- $ d = 2 $
- $ a_n = 1 + (n - 1) \cdot 2 = 1 + 2n - 2 = 2n - 1 $
✔ Answer: $ 2n - 1 $
---
- $ a = 10 $
- $ d = 4 $
- $ a_n = 10 + (n - 1) \cdot 4 = 10 + 4n - 4 = 4n + 6 $
✔ Answer: $ 4n + 6 $
---
- $ a = 2 $
- $ d = 5 $
- $ a_n = 2 + (n - 1) \cdot 5 = 2 + 5n - 5 = 5n - 3 $
✔ Answer: $ 5n - 3 $
---
- $ a = 3 $
- $ d = 6 $
- $ a_n = 3 + (n - 1) \cdot 6 = 3 + 6n - 6 = 6n - 3 $
✔ Answer: $ 6n - 3 $
---
- $ a = 11 $
- $ d = 31 - 11 = 20 $
- $ a_n = 11 + (n - 1) \cdot 20 = 11 + 20n - 20 = 20n - 9 $
✔ Answer: $ 20n - 9 $
---
- $ a = 20 $
- $ d = 3 $
- $ a_n = 20 + (n - 1) \cdot 3 = 20 + 3n - 3 = 3n + 17 $
✔ Answer: $ 3n + 17 $
---
- $ a = 1 $
- $ d = 6 $
- $ a_n = 1 + (n - 1) \cdot 6 = 1 + 6n - 6 = 6n - 5 $
✔ Answer: $ 6n - 5 $
---
- $ a = 100 $
- $ d = 25 $
- $ a_n = 100 + (n - 1) \cdot 25 = 100 + 25n - 25 = 25n + 75 $
✔ Answer: $ 25n + 75 $
---
- $ a = 13 $
- $ d = 9 $
- $ a_n = 13 + (n - 1) \cdot 9 = 13 + 9n - 9 = 9n + 4 $
✔ Answer: $ 9n + 4 $
---
- $ a = 1.5 $
- $ d = 0.5 $
- $ a_n = 1.5 + (n - 1) \cdot 0.5 = 1.5 + 0.5n - 0.5 = 0.5n + 1 $
✔ Answer: $ 0.5n + 1 $
(or $ \frac{1}{2}n + 1 $)
---
## ✔ Question 2: Find the nth term
These sequences are also arithmetic — let’s check differences.
- $ a = 10 $
- $ d = 7 - 10 = -3 $
- $ a_n = 10 + (n - 1)(-3) = 10 - 3n + 3 = -3n + 13 $
✔ Answer: $ -3n + 13 $
---
- $ a = 6 $
- $ d = -2 $
- $ a_n = 6 + (n - 1)(-2) = 6 - 2n + 2 = -2n + 8 $
✔ Answer: $ -2n + 8 $
---
- $ a = 9 $
- $ d = 4 - 9 = -5 $
- $ a_n = 9 + (n - 1)(-5) = 9 - 5n + 5 = -5n + 14 $
✔ Answer: $ -5n + 14 $
---
- $ a = 20 $
- $ d = -10 $
- $ a_n = 20 + (n - 1)(-10) = 20 - 10n + 10 = -10n + 30 $
✔ Answer: $ -10n + 30 $
---
- $ a = 5 $
- $ d = -1 - 5 = -6 $
- $ a_n = 5 + (n - 1)(-6) = 5 - 6n + 6 = -6n + 11 $
✔ Answer: $ -6n + 11 $
---
- $ a = 5 $
- $ d = -1 $
- $ a_n = 5 + (n - 1)(-1) = 5 - n + 1 = -n + 6 $
✔ Answer: $ -n + 6 $
---
- $ a = -6 $
- $ d = -13 - (-6) = -7 $
- $ a_n = -6 + (n - 1)(-7) = -6 -7n + 7 = -7n + 1 $
✔ Answer: $ -7n + 1 $
---
- $ a = -10 $
- $ d = -3 $
- $ a_n = -10 + (n - 1)(-3) = -10 -3n + 3 = -3n -7 $
✔ Answer: $ -3n - 7 $
---
- $ a = 2.5 $
- $ d = -0.5 $
- $ a_n = 2.5 + (n - 1)(-0.5) = 2.5 - 0.5n + 0.5 = -0.5n + 3 $
✔ Answer: $ -0.5n + 3 $
(or $ -\frac{1}{2}n + 3 $)
---
## ✔ Final Answers Summary
| Part | nth Term |
|------|----------|
| (a) | $ 3n + 2 $ |
| (b) | $ 5n + 4 $ |
| (c) | $ 2n - 1 $ |
| (d) | $ 4n + 6 $ |
| (e) | $ 5n - 3 $ |
| (f) | $ 6n - 3 $ |
| (g) | $ 20n - 9 $ |
| (h) | $ 3n + 17 $ |
| (i) | $ 6n - 5 $ |
| (j) | $ 25n + 75 $ |
| (k) | $ 9n + 4 $ |
| (l) | $ 0.5n + 1 $ |
---
| Part | nth Term |
|------|----------|
| (a) | $ -3n + 13 $ |
| (b) | $ -2n + 8 $ |
| (c) | $ -5n + 14 $ |
| (d) | $ -10n + 30 $ |
| (e) | $ -6n + 11 $ |
| (f) | $ -n + 6 $ |
| (g) | $ -7n + 1 $ |
| (h) | $ -3n - 7 $ |
| (i) | $ -0.5n + 3 $ |
---
Let me know if you'd like these verified with actual values or want help on non-linear sequences!
---
🔷 Key Concept: Arithmetic Sequence
For an arithmetic sequence, the nth term is given by:
$$
a_n = a + (n - 1)d
$$
Where:
- $ a $ = first term
- $ d $ = common difference (difference between consecutive terms)
- $ n $ = position in the sequence
We'll use this formula for all sequences where the pattern is linear (i.e., constant difference).
---
## ✔ Question 1: Find the nth term
(a) 5, 8, 11, 14, ...
- First term $ a = 5 $
- Common difference $ d = 8 - 5 = 3 $
- $ a_n = 5 + (n - 1) \cdot 3 = 5 + 3n - 3 = 3n + 2 $
✔ Answer: $ 3n + 2 $
---
(b) 9, 14, 19, 24, ...
- $ a = 9 $
- $ d = 14 - 9 = 5 $
- $ a_n = 9 + (n - 1) \cdot 5 = 9 + 5n - 5 = 5n + 4 $
✔ Answer: $ 5n + 4 $
---
(c) 1, 3, 5, 7, ...
- $ a = 1 $
- $ d = 2 $
- $ a_n = 1 + (n - 1) \cdot 2 = 1 + 2n - 2 = 2n - 1 $
✔ Answer: $ 2n - 1 $
---
(d) 10, 14, 18, 22, ...
- $ a = 10 $
- $ d = 4 $
- $ a_n = 10 + (n - 1) \cdot 4 = 10 + 4n - 4 = 4n + 6 $
✔ Answer: $ 4n + 6 $
---
(e) 2, 7, 12, 17, ...
- $ a = 2 $
- $ d = 5 $
- $ a_n = 2 + (n - 1) \cdot 5 = 2 + 5n - 5 = 5n - 3 $
✔ Answer: $ 5n - 3 $
---
(f) 3, 9, 15, 21, ...
- $ a = 3 $
- $ d = 6 $
- $ a_n = 3 + (n - 1) \cdot 6 = 3 + 6n - 6 = 6n - 3 $
✔ Answer: $ 6n - 3 $
---
(g) 11, 31, 51, 71, ...
- $ a = 11 $
- $ d = 31 - 11 = 20 $
- $ a_n = 11 + (n - 1) \cdot 20 = 11 + 20n - 20 = 20n - 9 $
✔ Answer: $ 20n - 9 $
---
(h) 20, 23, 26, 29, ...
- $ a = 20 $
- $ d = 3 $
- $ a_n = 20 + (n - 1) \cdot 3 = 20 + 3n - 3 = 3n + 17 $
✔ Answer: $ 3n + 17 $
---
(i) 1, 7, 13, 19, ...
- $ a = 1 $
- $ d = 6 $
- $ a_n = 1 + (n - 1) \cdot 6 = 1 + 6n - 6 = 6n - 5 $
✔ Answer: $ 6n - 5 $
---
(j) 100, 125, 150, 175, ...
- $ a = 100 $
- $ d = 25 $
- $ a_n = 100 + (n - 1) \cdot 25 = 100 + 25n - 25 = 25n + 75 $
✔ Answer: $ 25n + 75 $
---
(k) 13, 22, 31, 40, ...
- $ a = 13 $
- $ d = 9 $
- $ a_n = 13 + (n - 1) \cdot 9 = 13 + 9n - 9 = 9n + 4 $
✔ Answer: $ 9n + 4 $
---
(l) 1.5, 2, 2.5, 3, ...
- $ a = 1.5 $
- $ d = 0.5 $
- $ a_n = 1.5 + (n - 1) \cdot 0.5 = 1.5 + 0.5n - 0.5 = 0.5n + 1 $
✔ Answer: $ 0.5n + 1 $
(or $ \frac{1}{2}n + 1 $)
---
## ✔ Question 2: Find the nth term
These sequences are also arithmetic — let’s check differences.
(a) 10, 7, 4, 1, ...
- $ a = 10 $
- $ d = 7 - 10 = -3 $
- $ a_n = 10 + (n - 1)(-3) = 10 - 3n + 3 = -3n + 13 $
✔ Answer: $ -3n + 13 $
---
(b) 6, 4, 2, 0, ...
- $ a = 6 $
- $ d = -2 $
- $ a_n = 6 + (n - 1)(-2) = 6 - 2n + 2 = -2n + 8 $
✔ Answer: $ -2n + 8 $
---
(c) 9, 4, -1, -6, ...
- $ a = 9 $
- $ d = 4 - 9 = -5 $
- $ a_n = 9 + (n - 1)(-5) = 9 - 5n + 5 = -5n + 14 $
✔ Answer: $ -5n + 14 $
---
(d) 20, 10, 0, -10, ...
- $ a = 20 $
- $ d = -10 $
- $ a_n = 20 + (n - 1)(-10) = 20 - 10n + 10 = -10n + 30 $
✔ Answer: $ -10n + 30 $
---
(e) 5, -1, -7, -13, ...
- $ a = 5 $
- $ d = -1 - 5 = -6 $
- $ a_n = 5 + (n - 1)(-6) = 5 - 6n + 6 = -6n + 11 $
✔ Answer: $ -6n + 11 $
---
(f) 5, 4, 3, 2, ...
- $ a = 5 $
- $ d = -1 $
- $ a_n = 5 + (n - 1)(-1) = 5 - n + 1 = -n + 6 $
✔ Answer: $ -n + 6 $
---
(g) -6, -13, -20, -27, ...
- $ a = -6 $
- $ d = -13 - (-6) = -7 $
- $ a_n = -6 + (n - 1)(-7) = -6 -7n + 7 = -7n + 1 $
✔ Answer: $ -7n + 1 $
---
(h) -10, -13, -16, -19, ...
- $ a = -10 $
- $ d = -3 $
- $ a_n = -10 + (n - 1)(-3) = -10 -3n + 3 = -3n -7 $
✔ Answer: $ -3n - 7 $
---
(i) 2.5, 2, 1.5, 1, ...
- $ a = 2.5 $
- $ d = -0.5 $
- $ a_n = 2.5 + (n - 1)(-0.5) = 2.5 - 0.5n + 0.5 = -0.5n + 3 $
✔ Answer: $ -0.5n + 3 $
(or $ -\frac{1}{2}n + 3 $)
---
## ✔ Final Answers Summary
Question 1:
| Part | nth Term |
|------|----------|
| (a) | $ 3n + 2 $ |
| (b) | $ 5n + 4 $ |
| (c) | $ 2n - 1 $ |
| (d) | $ 4n + 6 $ |
| (e) | $ 5n - 3 $ |
| (f) | $ 6n - 3 $ |
| (g) | $ 20n - 9 $ |
| (h) | $ 3n + 17 $ |
| (i) | $ 6n - 5 $ |
| (j) | $ 25n + 75 $ |
| (k) | $ 9n + 4 $ |
| (l) | $ 0.5n + 1 $ |
---
Question 2:
| Part | nth Term |
|------|----------|
| (a) | $ -3n + 13 $ |
| (b) | $ -2n + 8 $ |
| (c) | $ -5n + 14 $ |
| (d) | $ -10n + 30 $ |
| (e) | $ -6n + 11 $ |
| (f) | $ -n + 6 $ |
| (g) | $ -7n + 1 $ |
| (h) | $ -3n - 7 $ |
| (i) | $ -0.5n + 3 $ |
---
Let me know if you'd like these verified with actual values or want help on non-linear sequences!
Parent Tip: Review the logic above to help your child master the concept of sequences worksheet nth.
