√ Arithmetic Series Formulas, Sequences, Forms, Example Questions & Answers
Arithmetic Series Formulas, Sequences, Forms, Example Questions & Answers - On this occasion About Knowledge We will discuss about Arithmetic Series. Which in this discussion explains various kinds of problems regarding the arithmetic series formula. For more details, let's look at the following review.
Arithmetic Series Formulas, Sequences, Forms, Example Questions & Answers
A number sequence is a set of numbers ordered according to a certain rule/pattern connected by a "," sign. If the "," sign is replaced by a "+" sign, then it is called a series. Each of these numbers is called the terms of the sequence
Definition of Arithmetic
Arithmetic or arithmetic is a word that comes from the Greek αριθμός = numbers which used to be called Arithmetic. Arithmetic is the oldest branch (or precursor) of mathematics that studies the basic operations on numbers.
Arithmetic Sequence
An Arithmetic Sequence is a sequence of numbers with a certain pattern in the form of sums that have the same/fixed difference or difference.
Arithmetic Sequence Formula
The terms are expressed by the following formula:
U1, U2, U3, ….Un
a, a+ b, a+2b, a + 3b, …., a + (n-1) b
The difference (difference) is expressed by b
b = U2 – U1 = U3 – U2 = Un – Un – 1
The nth term of an arithmetic sequence (Un) is expressed by the formula:
Un = a + (n-1) b
Information :
Un = the nth term with n = 1,2,3, …
a = first term → U1 = a
b = difference/difference
(1) 3, 7, 11, 15, 19, …
(2) 30, 25, 20, 15, 10,…
Forms of Arithmetic Sequences
In this case, it is necessary to pay attention to some information on the formula for the form of an arithmetic sequence as follows:
a, (a+b), (a+2b), (a+3b), ….., (a+(n-1)b)
Formula:
b = Un – Un-1
nth tribe :
un = a + (n-1)b
Or
un = Sn – Sn-1
Information:
a = U1 = First term
b = different
n = many terms
Un= The nth term
Example of an Arithmetic Sequence
The first term of the arithmetic sequence is 3 and the difference = 4, the 10th term of the arithmetic sequence is…
Completion:
a = 3
b = 4
un = a + (n-1) b
u10 = 3 + (10-1)4
= 3 + 36
= 39
Examples of Arithmetic Sequence Problems
Determine the 15th term of the sequence 2, 6, 10, 14, ...
Answer:
n=15
b = 6-2 = 10 – 6 = 4
U1 = a = 2
un = a + (n-1) b
u15 = 2 + (15-1)4
= 2 + 14.4
= 2 + 56 = 58
Derive the Formula for the nth Element of an Arithmetic Sequence
If U1 = a, U2, U3,…, Un,… is an arithmetic sequence, then the nth element of the sequence can be derived in the following way.
U1 = a
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
?
Un = a + (n-1)b
So the general formula for the nth element of an arithmetic sequence with the first element a and difference b is:
Un = a + (n-1)b
Example Question 1
It is known that the arithmetic sequence with the 2nd element is 10 and difference = 2. Determine the 7th element of the sequence.
Completion:
It is known that U2 = 10, b = 2. By using the formula Un = a + (n-1)b, it is obtained
U2 = a + (2-1)b
U2 = a + b
a = U2 – b
= 10 – 2
= 8.
U7 = a + (7-1) b
= a + 6 b
= 8 + 6 (2)
= 8 + 12
= 20.
So the 7th element of the sequence is 20.
Example Problem 2
Starting in 2000, Pak Arman owned a sugar cane plantation. Arman's sugar cane plantation income at the end of 2000 was IDR 6,000,000. Starting in 2001, Pak Arman fertilized his sugar cane plantation with manure. Pak Arman estimates that at the end of each year, his sugar cane plantation income increases by IDR 500,000. What is the estimated income of Pak Arman's sugar cane plantation at the end of 2005?
Completion:
For example:
a = income of Pak Arman's sugar cane plantation at the end of 2000.
b = estimated increase in income from Mr. Arman's sugar cane plantation at the end of each year.
P 2005 = estimated income from Pak Arman's plantation at the end of 2005.
So the formula is determined, a = Rp. 6,000,000.-, b = Rp. 500,000.-, and P2005 which will be sought.
Because the estimated increase in income from Mr. Arman's sugar cane plantation at the end of each year is the same. So to determine the income of Pak Arman's plantation at the end of 2005. We can apply the formula for the nth element of an arithmetic sequence by
U1 = a = a = IDR 6,000,000, b = IDR 500,000.
P2005 = U6 = a + 5b
= 6.000.000 + 5(500.000)
= 6.000.000 + 2.500.000
= 8.500.000.
So, the estimated income of Pak Arman's sugar cane plantation at the end of 2005 was IDR 8,500,000. With an arithmetic series, we can form a sequence related to the series. Such a sequence is called an arithmetic sequence.
Example Problem 3
Find the sum of all the odd numbers between 50 and 100.
Completion:
It is known that a = 51, b = 2, and Un = 99.
To find the sum of all the odd numbers between 50 and 100, we first find the number of odd numbers between 50 and 100, namely n. By using the formula:
Un = a + (n – 1) b
99 = 51 + (n – 1)(2)
99 = 51 + 2n – 2
99 = 49 + 2n
2n = 99 – 49
n=25.
Furthermore, with the formula for the sum of the first n terms of an arithmetic sequence,
sn =
1
2
n[2a + (n -1)b]
obtained:
S25 =
1
2
(25)[2(51) + (25 -1)(2)]
= 25(51 + 24)
= 25(75)
= 1.875.
So the sum of all the odd numbers between 50 and 100 is 1.875.
Thus our explanation this time about Arithmetic Series Formulas, Sequences, Forms, Example Questions & Answers. Hopefully it can be useful and add knowledge to all of us.
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