**Arithmetic Sequences and Series :**

An arithmetic sequence is a sequence in which the difference between each consecutive term is constant.

An arithmetic series is the sum of the terms of an arithmetic sequence.

**Example : **

Is the sequence arithmetic ? If so, what is the common difference ? What is the next term in the sequence ?

3, 8, 13, 18, 23,.............

**Solution :**

This is a sequence, a function whose domain is the Natural numbers.

Create a table that shows the term number, or domain, and the term, or range.

An arithmetic sequence is a sequence with a constant difference between consecutive terms. The difference is known as the common difference, or d.

This sequence is an arithmetic sequence with common difference,

d = 5

The next term in the sequence is

23 + 5, or 28

In an arithmetic sequence, each term can be represented by f(n) where n represents the number of a particular term.

Let us consider an arithmetic sequence where the first term is 3 and the common difference is 5.

So, for n = 1,

f(1) = 3

If n > 1, each term is the sum of the previous term and the common difference 5.

f(2) = f(1) + 5

f(3) = f(2) + 5

f(4) = f(3) + 5

In this way, we have

f(n) = f(n - 1) + 5

Write the general rule for an arithmetic sequence as a piecewise-defined function :

This is the recursive definition for an arithmetic sequence. Each term is defined by operations on the previous term.

Another way to write recursive definition for an arithmetic sequence is

In the notation shown above, the subscript shows the number of the term.

**Example : **

Is the sequence 7, 10, 13, 16, ........... arithmetic ? If so, write the recursive definition for the sequence.

**Solution : **

In the given sequence, difference between any two consecutive terms along the sequence is 3.

So, this is an arithmetic sequence.

The recursive definition for this sequence is

**Example 1 : **

Given the recursive definition :

What is an explicit definition for the sequence ?

**Solution : **

An explicit definition, also written as

a_{n} = a_{1} + d(n - 1)

allows us to find any term in the sequence without knowing the previous term.

Use the recursive definition to find a pattern :

a_{1} = 3 = 3 + 0(5)

a_{2} = a_{1} + 5 = 3 + 5 = 3 + 1(5)

a_{3} = a_{2} + 5 = 8 + 5 = [3 + 5] + 5 = 3 + 2(5)

a_{4} = a_{3} + 5 = 13 + 5 = [3 + 5 + 5] + 5 = 3 + 3(5)

So, the explicit form is

a_{n} = 3 + (n - 1)(5)

In general, the explicit definition of an arithmetic sequence is

a_{n} = a_{1} + d(n - 1)

**Example 2 : **

Given the explicit definition :

a_{n} = 16 + 3(n - 1)

What is the recursive definition for the arithmetic sequence ?

**Solution : **

Comparing a_{n} = a_{1} + d(n - 1) and a_{n} = 16 + 3(n - 1), we get

the common difference d = 3 and a_{1} = 16

Hence, the recursive definition is

**Example :**

A high school auditorium has 20 seats in the first row and 35 seats in the sixth row. The number of seats in each row forms an arithmetic sequence.

A. What is the explicit definition for the sequence ?

B. How many seats are in the thirteenth row ?

**Solution (A) : **

The problem states that

a_{1} = 20, n = 6 and a_{6} = 35

Write the general explicit formula.

a_{n} = a_{1} + d(n - 1)

Substitute.

35 = 20 + d(6 - 1)

Simplify.

35 = 20 + 5d

Subtract 20 from each side.

15 = 5d

Divide each side by 5.

3 = d

So, each row has two more seats than the previous row.

Then, the explicit definition is

a_{n} = 20 + 3(n - 1)

**Solution (B) : **

Write the explicit formula.

a_{n} = 20 + 3(n - 1)

We have to find the number of seats in thirteenth row. So we have substitute 13 for n.

a_{13} = 20 + 3(13 - 1)

Simplify.

a_{13} = 20 + 3(12)

a_{13} = 20 + 36

a_{13} = 56

Hence, there are 56 seats in thirteenth row.

**Example 1 : **

Find the sum of the terms in the arithmetic sequence :

5, 9, 13, 17, 21

What is the general formula for an arithmetic series ?

**Solution :**

A finite series is the sum of the terms in a finite sequence. A finite arithmetic series is the sum of the terms in an arithmetic sequence. For the sum of n numbers in a sequence, we can use recursive formula or simply add the terms.

Sn = + a_{1} + a_{2 }+ a_{3 }+ a_{4 }+ ...................... + a_{n}

(This represents a partial sum of a series, because it is the sum of a finite number of terms, n, in the series)

5 + 9 + 13 + 17 + 21 = 65

To find the sum of a series with many terms, we can use an explicit definition.

**Step 1 : **

To find the explicit definition for the sum, use the Commutative Property of Addition and reverse the order of the terms in the recursive series.

S_{5} = 21 + 17 + 13 + 9 + 5

**Step 2 : **

Add the two expressions for the series, so we are adding the first term to last term and the second term to the second-to-last term, and so on.

**Step 3 : **

Simplify.

2 ⋅ S_{5} = 5(26)

2 ⋅ S_{5} = 5(5 + 21)

Divide each side by 2.

S_{5} = 5(5 + 21) / 2

**Step 4 : **

Write the general formula.

S_{n} = n(a_{1} + a_{n}) / 2

**Example 2 : **

Find the sum of the terms in the arithmetic sequence :

4, 10, 16, 22, ..................., 286

**Solution : **

**Step 1 :**

In the given sequence, we have

a_{1} = 4, a_{n} = 286 and d = 6

**Step 2 : **

Find the value of n.

n = [(a_{n }- a_{1}) / d] + 1

Substitute.

n = [(286 - 4) / 6] + 1

Simplify.

n = 282 / 6 + 1

n = 47 + 1

n = 48

**Step 3 :**

Write the general formula to find the sum.

S_{n} = n(a_{1} + a_{n}) / 2

Substitute.

S_{48} = 48(4 + 286) / 2

Simplify.

S_{48} = 48(290) / 2

S_{48} = 48(290) / 2

S_{48} = 6960

Hence, the sum of the terms in the given arithmetic sequence is 6960.

After having gone through the stuff given above, we hope that the students would have understood, "Arithmetic Sequences and Series".

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