Complete Arithmetic Sequences and Series Materials

Math formula – In the discussion of mathematical formulas this time, we will study a material that can be said to be easy and difficult, namely arithmetic sequences and series. Here I will start the discussion by offering an understanding of arithmetic sequences and series, then after that I will invite you to understand how to apply these formulas. I hope that after you study this material, you will be able to answer questions related to arithmetic sequences and series more quickly and quickly. Therefore, pay attention to each of the clarifications below with focus and concentration.

In the discussion of mathematical formulas this time the Complete Arithmetic Sequences and Series Materials

Complete Discussion of Arithmetic Sequences and Series Materials

What is Arithmetic Sequence?

Before understanding the meaning of an arithmetic sequence, we must first understand the meaning of a sequence of numbers. A number sequence is a sequence of numbers made according to certain rules. While the arithmetic sequence can be defined as a sequence of numbers in which each successive pair of terms contains the exact same difference value, for example, the sequence of numbers: 2, 4, 6, 8, 10, 12, 14, …

This sequence of numbers can be called an arithmetic sequence because each term has the same difference, which is 2. The difference value that appears in an arithmetic sequence is usually denoted by using letters b. Each number that forms the sequence of an arithmetic sequence is called a term. The nth term of an arithmetic sequence can be symbolized by the symbol Un so to write the 3rd term of a sequence we can write U3. However, there is a special exception for the first term in a number sequence, the first term is symbolized by using the character a.

Also read: Mathematical Formulas of Exponent and Logarithmic Functions Class 12

So, in general, an arithmetic series has the form:

U1,U2,U3,U4,U5,…Un-1
a, atb, a+2b, a+3b, a+4b,…a+(n-1)b

How to Determine the Formula of the nth Term of a Sequence

In arithmetic sequences, finding the formula for the nth term is simpler because they have the same difference value, so the formula is:

U2 = a + b
U3 = u2 + b = (a + b) + b = a + 2b
U4 = u3 + b = (a + 2b) + b = a + 3b
U5 = u4 + b = (a + 3b) + b = a + 4b
U6 = u5 + b = (a + 4b) + b = a + 5b
U7 = u6 + b = (a + 5b) + b = a + 6b.

U68 = u67+b = (a + 66b) + b = a + 67b
U87 = u86+b = (a + 85b) + b = a + 86b

Based on the sequence pattern above, we can conclude that nth formula of an arithmetic sequence is:

Un = a + (n – 1)b where n is natural number

Definition of Arithmetic Series

An arithmetic series can be defined as the sum total of the elements of an arithmetic sequence that are calculated sequentially. For reference we take an arithmetic sequence 8,12,16,20,24 then the arithmetic series is 8+12+16+20+24

To calculate the arithmetic series is still fairly simple because the number of terms is still small:

8+12+16+20+24 = 80

However, imagine if the arithmetic series consisted of hundreds of terms, of course it would be difficult to calculate it, right? Therefore, we must know the formula for calculating the sum of an arithmetic series. Commonly used formulas are:

Sn = (a + Un) × n : 2

Previously we already know the formula for calculating Un, then the formula can be modified to:

Sn = (a + a + (n – 1)b) × n : 2

Inserts in Arithmetic Series

Inserts in an arithmetic series can be obtained by adding another small arithmetic series between two successive terms in an arithmetic series. To understand it more simply consider the following references:

Initial arithmetic series: 2+8+14+20+26+32
Arithmetic series after insertion: 2+4+6+8+10+12+14++16+18+20+22+24+26+28+30+32

The value of the difference in an arithmetic series that has been inserted (b1) can be known by using the formula:

b1 = b/(k+1)

b1 = difference in the inserted series
b = difference in the initial arithmetic series
k = number of inserted numbers

as a reference for calculating the difference in the new series in the arithmetic series that I have written above are:

Starting line: 2+8+14+20+26+32
New series: 2+4+6+8+10+12+14++16+18+20+22+24+26+28+30+32

Formula: b1 = b/(k+1)

Is known:

b = 8 – 2 = 6
k = 2

So:
b1 = 6/(2+1)
b1 = 6/3
b1 = 2

This is a clarification of the meaning of arithmetic sequences and series. Actually, this material is not too difficult to learn, we just have to be more careful and careful in calculating every tribe that exists so that it turns out to be true. To deepen your understanding of arithmetic sequences and series, you should continue to practice by trying to solve problems related to the material above.