What is Arithmetic Sequence

What is Arithmetic Sequence

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Arithmetic Sequences Definition

A list of numbers with a clear pattern is known as an arithmetic sequence. We can tell if a series of numbers is arithmetic by taking any number and subtracting it by the number before it. In an arithmetic sequence, the difference between two numbers stays the same.

The letter d stands for the common difference, the constant difference between the numbers in an arithmetic sequence. We have an increasing arithmetic sequence if d is positive. We have a decreasing arithmetic sequence if d is negative.

Arithmetic Sequences Formula

We can use the following formula to find different terms for the arithmetic sequence:

an = a1 + d (n  1), where:

  • a1: The first term
  • an: The nth term
  • n: The term poistion
  • d: Common difference

How to figure out the terms in an arithmetic sequence?

To find any term in the arithmetic sequence, we need to know a common difference, the position of the term we want to find, and a term in the sequence.

Example

Consider the arithmetic sequence: 3, 7, 11, 15, 19, ..., Find a28 = ?

Solution

Before using the arithmetic sequence formula, we need to know the first term, the common difference, and the position of the term we want to find:

  • The first term: a1 = 3
  • common difference: d = 4
  • Term's position: n = 28

Now, we put these numbers into the formula and find the answer:

an = a1 + d (n  1)  a28 = 3 + 4(28  1)  a28 = 111

Summing an Arithmetic Series

To summing the terms in an arithmetic sequence, use this formula:

n  1k = 0 (a + kd) = n2(2a + (n  1)d)

Example

Find the sum of the first 12 terms for the sequence: 1, 6, 11, 16, 21, ...

Solution

  • The first term: a1 = 1
  • common difference: d = 5
  • n = 12

So:

n  1k = 0 (a + kd) = n2(2a + (n  1)d) 

11k = 0 (1 + 5k) = 122(2 + (11)5) = 6×(57) = 342

Free printable Worksheets

Exercises for Arithmetic Sequences

1) Find the explicit formula: 5, 14, 23, 32, 41, ...

2) Find the explicit formula: 7, 2, 3, 8, 13, ...

3) Find the explicit formula and a12: 22, 16, 10, 4, 2, ...

4) Find the explicit formula and a21: a30 = 72, d = 5

5) Find the first 10 terms: a15 = 50, d = 4.5

6) Find the first 7 terms: a41 = 178, d = 4

7) Find a3045, , 41.8, 38.6, 35.4, ...

8) Find a27: a41 = 167, d = 6.4

9) Find the explicit formula and the sum of the first five terms: a1 = 10, d = 3

10) Find the explicit formula and the sum of the first 16 terms: 55, 48, 41, 34, 27, ...

 

1) Find the explicit formula: 5, 14, 23, 32, 41, ...

a1 = 5, d = 14  5 = 9, an = a1 + d (n  1)
 an = 5 + 9 (n  1)

2) Find the explicit formula: 7, 2, 3, 8, 13, ...

a1 = 7, d = 2  (7) = 5, an = a1 + d (n  1)
 an = 7 + 5 (n  1)

3) Find the explicit formula and a12: 22, 16, 10, 4, 2, ...

a1 = 22, d = 16  (22) = 6, an = a1 + d (n  1)
 an = 22 + 6 (n  1)  a12 = 22 + 6 (12  1) = 44

4) Find the explicit formula and a21: a30 = 72, d = 5

a30 = 72, d = 5, an = a1 + d (n  1)
 a30 = a1 + (5) (30  1)  a1 = a30 + 5(29) = 72 + 145 = 217
 an = 217  5 (n  1)  a21 = 217  5(20) = 117

5) Find the first 10 terms: a15 = 50, d = 4.5

a15 = 50, d = 4.5, an = a1 + d (n  1)
 a15 = a1 + 4.5 (15  1)  a1 = 50  4.5(14) = 13
13, 8.5, 4, 0.5, 5, 9.5, 14, 18.5, 23, 27.5, ...

6) Find the first 7 terms: a41 = 178, d = 4

a41 = 178, d = 4, an = a1 + d (n  1)
 a41 = a1 + 4 (41  1)  a1 = 178  4(40) = 18
18, 22, 26, 30, 34, 38, 42, ...

7) Find a3045, , 41.8, 38.6, 35.4, ...

a1 = 45, d = 41.8  45 = 3.2, an = a1 + d (n  1)
 an = 45  3.2 (n  1)  a30 = 45  3.2 (30  1) = 47.8

8) Find a27: a41 = 167, d = 6.4

a41 = 167, d = 6.4, an = a1 + d (n  1)
 a41 = a1 + 6.4 (41  1)  a1 = 167  6.4(40) = 98
 an = 98 + 6.4 (n  1)  a27 = 98 + 6.4(26) = 68.4

9) Find the explicit formula and the sum of the first five terms: a1 = 10, d = 3

a1 = 10, d = 3, an = a1 + d (n  1)
 an = 10 + 3 (n  1)

n  1k = 0 (a + kd) = n2 (2a + (n  1)d)

 4k = 0 (10 + 3k) = 52 (20 + (4)3) = 52×(32) = 80

10) Find the explicit formula and the sum of the first 16 terms: 55, 48, 41, 34, 27, ...

a1 = 55, d = 48  (55) = 7, an = a1 + d (n  1)
 an = 55 + 7 (n  1)

n  1k = 0 (a + kd) = n2 (2a + (n  1)d)

 15k = 0 (55 + 7k) = 162 (110 + (15)7) = 8×(5) = 40

Arithmetic Sequences Practice Quiz