SL 1.2: Arithmetic sequences and series.

Consider the following sequences, 

Oh, firstly, what is a sequence? 

Sequence:

A sequence represents numbers formed in succession and arranged in a fixed order defined by a certain rule.

Now consider the following sequences,

1,2,3,4,5,...

2,4,6,8,10,... 

5,10,15,20,25,... 

7,5,3,-1,... 

There is something common in the above sequences, if you check the difference between any two consecutive terms of any sequence, it's constant! 

A sequence of numbers in which the difference between consecutive terms is constant is called an Arithmetic sequence ( Or Arithmetic Progression).

Arithmetic Sequence

The general form of an arithmetic sequence can be written as:

• First term: $a_1$
• Second term: $a_2 = a_1 + d$
  
• Third term: $a_3 = a_1 + 2d$
  
• $n$-th term: $a_n = a_1 + (n-1)d$.

Common Difference

The common difference $d$ is calculated as: $d = a_{n+1} - a_n$

Arithmetic Series

An arithmetic series is the sum of the terms of an arithmetic sequence. The sum of the first $n$ terms, denoted as $S_n$, can be calculated using the formula:

$$S_n = \frac{n}{2} \times (a_1 + a_n)$$

Alternatively, you can use:

$$S_n = \frac{n}{2} \times (2a_1 + (n-1)d)$$

Example

Finding Terms in an Arithmetic Sequence: Suppose $a_1 = 3$ and $d = 2$. The first few terms would be:

• $a_1 = 3$
  
• $a_2 = 3 + 2 = 5$
  
• $a_3 = 5 + 2 = 7$
  
• $a_4 = 7 + 2 = 9$
  
• and so on...

Calculating the Sum: If you want to find the sum of the first 5 terms of this sequence:

1. Identify the 5th term: $a_5=a_1+4d=3+4\times 2=11$
2. Use the sum formula: $S_5 = \frac{5}{2} \times (3 + 11) = \frac{5}{2} \times 14 = 35$
   

More Examples

1. In an arithmetic series, the first term is −7 and the sum of the first 20 terms is 620.

    Find the common difference.

     a.

b. Find the value of the 78th term.

Solution: 

The formula for the sum of the first $n$ terms of an arithmetic series is:

$$S_n = \frac{n}{2} \times (2a_1 + (n-1)d)$$

For our case:

$$S_{20} = \frac{20}{2} \times (2 \cdot -7 + (20-1)d)$$

Substituting the values:

$$620 = 10 \times (2 \cdot -7 + 19d)$$

                                                                d = 4 ( using GDC ) 

So, the common difference $d$ is 4.

b. Finding the 78th Term

The formula for the $n$-th term of an arithmetic sequence is:

$$a_n = a_1 + (n-1)d$$

To find the 78th term $a_{78}$:

$$a_{78} = -7 + (78-1) \cdot 4$$

$$a_{78} = 301$$

2. The first three terms of an arithmetic sequence are 5, 6.7, 8.4.

a. Find the common difference.

Solution: 

We can calculate the difference between consecutive terms to find the common difference in the arithmetic sequence where the first three terms are 5, 6.7, and 8.4.

The common difference $d$ can be found as follows:

1. From the first term to the second term:

   $$d = 6.7 - 5 = 1.7$$

2. From the second term to the third term:

   $$d = 8.4 - 6.7 = 1.7$$
   

Since both calculations yield the same result, the common difference $d$ is 1.7.

b. Find the 28th term of the sequence.

Solution: 

we can use the formula for the $n$-th term of an arithmetic sequence:

$$a_n = a_1 + (n-1)d$$

where $a_1$ is the first term, $d$ is the common difference, and $n$ is the term number.

For the 28th term ($n = 28$):

$$a_{28} = 5 + (28 - 1) \cdot 1.7$$

$$a_{28} = 50.9$$

Thus, the 28th term of the sequence is 50.9.

 

c. Find the sum of the first 28 terms.

Solution: 

The formula for the sum of the first $n$ terms of an A.P. is:

$$S_n = \frac{n}{2} \times (a_1 + a_n)$$

where $a_n$ is the $n$-th term.

For $n = 28$

$$S_{28} = \frac{28}{2} \times (a_1 + a_{28})$$

$$S_{28} = 14 \times (5 + 50.9)$$

$${\mathrm{ S}}_{28}=14\times 55.9=782.6$$

Thus, the sum of the first 28 terms $S_{28}$ is 782.6.

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