11+ Maths: Write variable expressions for arithmetic sequences

Introduction

Today, we are going to learn about an exciting topic in mathematics called ‘Arithmetic Sequences’.

Don’t worry! It might sound a bit complex, but trust me, it’s fun and quite simple when you get the hang of it, especially when we start writing variable expressions for them.

What is an Arithmetic Sequence?

An Arithmetic Sequence is a sequence of numbers in which the difference between any two consecutive terms is always the same. This constant difference is often called the ‘common difference’.

For example, consider the sequence 2, 4, 6, 8, 10. Here, the common difference is 2 (4-2 = 2, 6-4 = 2, and so on).

Writing Variable Expressions for Arithmetic Sequences

Now, to write a variable expression for an arithmetic sequence, we need to know the first term (let’s call it ‘a’) and the common difference (let’s call it ‘d’).

The general form of an arithmetic sequence is:

$$a, a + d, a + 2d, a + 3d, a + 4d, \dots$$

For example, in the sequence 2, 4, 6, 8, 10, the first term ‘a’ is 2 and the common difference ‘d’ is 2.

So, the variable expression for the nth term of this sequence would be:

$$a + (n – 1) * d$$

In our example, this becomes:

$$2 + (n – 1) * 2$$

This is the variable expression for the nth term of the sequence. Isn’t that neat?

Tips and Tricks

1. Always remember, in an arithmetic sequence, the difference between any two consecutive terms is constant.
2. The variable expression for the nth term of an arithmetic sequence is a + (n – 1) * d, where ‘a’ is the first term and ‘d’ is the common difference.
3. Be careful with your calculations. Even a small mistake can lead to a wrong answer.

Questions

Easy Level

1. Write a variable expression for the arithmetic sequence starting with 3 and a common difference of 2.
2. Write a variable expression for the arithmetic sequence starting with 5 and a common difference of 3.
3. Write a variable expression for the arithmetic sequence starting with 1 and a common difference of 4.
4. Write a variable expression for the arithmetic sequence starting with 2 and a common difference of 5.
5. Write a variable expression for the arithmetic sequence starting with 4 and a common difference of 1.
6. Write a variable expression for the arithmetic sequence starting with 6 and a common difference of 2.
7. Write a variable expression for the arithmetic sequence starting with 7 and a common difference of 3.
8. Write a variable expression for the arithmetic sequence starting with 8 and a common difference of 4.
9. Write a variable expression for the arithmetic sequence starting with 9 and a common difference of 1.
10. Write a variable expression for the arithmetic sequence starting with 10 and a common difference of 2.
11. Write a variable expression for the arithmetic sequence starting with 11 and a common difference of 3.
12. Write a variable expression for the arithmetic sequence starting with 12 and a common difference of 4.
13. Write a variable expression for the arithmetic sequence starting with 13 and a common difference of 1.
14. Write a variable expression for the arithmetic sequence starting with 14 and a common difference of 2.
15. Write a variable expression for the arithmetic sequence starting with 15 and a common difference of 3.
16. Write a variable expression for the arithmetic sequence starting with 16 and a common difference of 4.
17. Write a variable expression for the arithmetic sequence starting with 17 and a common difference of 1.
18. Write a variable expression for the arithmetic sequence starting with 18 and a common difference of 2.
19. Write a variable expression for the arithmetic sequence starting with 19 and a common difference of 3.
20. Write a variable expression for the arithmetic sequence starting with 20 and a common difference of 4.

Medium Level

1. Write a variable expression for the arithmetic sequence starting with 21 and a common difference of 5.
2. Write a variable expression for the arithmetic sequence starting with 22 and a common difference of 6.
3. Write a variable expression for the arithmetic sequence starting with 23 and a common difference of 7.
4. Write a variable expression for the arithmetic sequence starting with 24 and a common difference of 8.
5. Write a variable expression for the arithmetic sequence starting with 25 and a common difference of 9.
6. Write a variable expression for the arithmetic sequence starting with 26 and a common difference of 10.
7. Write a variable expression for the arithmetic sequence starting with 27 and a common difference of 11.
8. Write a variable expression for the arithmetic sequence starting with 28 and a common difference of 12.
9. Write a variable expression for the arithmetic sequence starting with 29 and a common difference of 13.
10. Write a variable expression for the arithmetic sequence starting with 30 and a common difference of 14.
11. Write a variable expression for the arithmetic sequence starting with 31 and a common difference of 15.
12. Write a variable expression for the arithmetic sequence starting with 32 and a common difference of 16.
13. Write a variable expression for the arithmetic sequence starting with 33 and a common difference of 17.
14. Write a variable expression for the arithmetic sequence starting with 34 and a common difference of 18.
15. Write a variable expression for the arithmetic sequence starting with 35 and a common difference of 19.
16. Write a variable expression for the arithmetic sequence starting with 36 and a common difference of 20.
17. Write a variable expression for the arithmetic sequence starting with 37 and a common difference of 21.
18. Write a variable expression for the arithmetic sequence starting with 38 and a common difference of 22.
19. Write a variable expression for the arithmetic sequence starting with 39 and a common difference of 23.
20. Write a variable expression for the arithmetic sequence starting with 40 and a common difference of 24.

Hard Level

1. Write a variable expression for the arithmetic sequence starting with 41 and a common difference of 25.
2. Write a variable expression for the arithmetic sequence starting with 42 and a common difference of 26.
3. Write a variable expression for the arithmetic sequence starting with 43 and a common difference of 27.
4. Write a variable expression for the arithmetic sequence starting with 44 and a common difference of 28.
5. Write a variable expression for the arithmetic sequence starting with 45 and a common difference of 29.
6. Write a variable expression for the arithmetic sequence starting with 46 and a common difference of 30.
7. Write a variable expression for the arithmetic sequence starting with 47 and a common difference of 31.
8. Write a variable expression for the arithmetic sequence starting with 48 and a common difference of 32.
9. Write a variable expression for the arithmetic sequence starting with 49 and a common difference of 33.
10. Write a variable expression for the arithmetic sequence starting with 50 and a common difference of 34.
11. Write a variable expression for the arithmetic sequence starting with 51 and a common difference of 35.
12. Write a variable expression for the arithmetic sequence starting with 52 and a common difference of 36.
13. Write a variable expression for the arithmetic sequence starting with 53 and a common difference of 37.
14. Write a variable expression for the arithmetic sequence starting with 54 and a common difference of 38.
15. Write a variable expression for the arithmetic sequence starting with 55 and a common difference of 39.
16. Write a variable expression for the arithmetic sequence starting with 56 and a common difference of 40.
17. Write a variable expression for the arithmetic sequence starting with 57 and a common difference of 41.
18. Write a variable expression for the arithmetic sequence starting with 58 and a common difference of 42.
19. Write a variable expression for the arithmetic sequence starting with 59 and a common difference of 43.
20. Write a variable expression for the arithmetic sequence starting with 60 and a common difference of 44.

Answers

The answers for all the questions are based on the formula for the nth term of an arithmetic sequence, which is:

$$a + (n – 1) * d$$

where ‘a’ is the first term and ‘d’ is the common difference.

Here are the answers for the easy, medium and hard level questions:

Easy Level

1. 3 + (n – 1) * 2
2. 5 + (n – 1) * 3
3. 1 + (n – 1) * 4
4. 2 + (n – 1) * 5
5. 4 + (n – 1) * 1
6. 6 + (n – 1) * 2
7. 7 + (n – 1) * 3
8. 8 + (n – 1) * 4
9. 9 + (n – 1) * 1
10. 10 + (n – 1) * 2
11. 11 + (n – 1) * 3
12. 12 + (n – 1) * 4
13. 13 + (n – 1) * 1
14. 14 + (n – 1) * 2
15. 15 + (n – 1) * 3
16. 16 + (n – 1) * 4
17. 17 + (n – 1) * 1
18. 18 + (n – 1) * 2
19. 19 + (n – 1) * 3
20. 20 + (n – 1) * 4

Medium Level

1. 21 + (n – 1) * 5
2. 22 + (n – 1) * 6
3. 23 + (n – 1) * 7
4. 24 + (n – 1) * 8
5. 25 + (n – 1) * 9
6. 26 + (n – 1) * 10
7. 27 + (n – 1) * 11
8. 28 + (n – 1) * 12
9. 29 + (n – 1) * 13
10. 30 + (n – 1) * 14
11. 31 + (n – 1) * 15
12. 32 + (n – 1) * 16
13. 33 + (n – 1) * 17
14. 34 + (n – 1) * 18
15. 35 + (n – 1) * 19
16. 36 + (n – 1) * 20
17. 37 + (n – 1) * 21
18. 38 + (n – 1) * 22
19. 39 + (n – 1) * 23
20. 40 + (n – 1) * 24

Hard Level

1. 41 + (n – 1) * 25
2. 42 + (n – 1) * 26
3. 43 + (n – 1) * 27
4. 44 + (n – 1) * 28
5. 45 + (n – 1) * 29
6. 46 + (n – 1) * 30
7. 47 + (n – 1) * 31
8. 48 + (n – 1) * 32
9. 49 + (n – 1) * 33
10. 50 + (n – 1) * 34
11. 51 + (n – 1) * 35
12. 52 + (n – 1) * 36
13. 53 + (n – 1) * 37
14. 54 + (n – 1) * 38
15. 55 + (n – 1) * 39
16. 56 + (n – 1) * 40
17. 57 + (n – 1) * 41
18. 58 + (n – 1) * 42
19. 59 + (n – 1) * 43
20. 60 + (n – 1) * 44