This article will discuss arithmetic and geometry formulas along with examples of questions and their discussion.
Arithmetic and geometry are topics that we study in school in mathematics. Where when national examinations, college entrance exams or exams in applying for jobs often have questions about both of them.
Not only that, we must also be able to distinguish between arithmetic sequence and series or geometric sequence and series, because rules and sequences can make it easier we are in the completion of the calculation, such as bank interest, increase in production, and profit / loss of a business. Well, let's look at the following description!
Arithmetic Formulas
Look at the table above, observe how the difference in numbers from our left to our right? And then, observe what is the difference from top to bottom?
From the results of the observation, we find that the difference from one line is always fixed. Suppose that from our right row to our left: 1, 2, 3, 4, 5 (difference 1).
5 – 1 = 4; 4 – 1 = 3; 3 – 1 = 2; 2 – 1 = 1
And from the top row down: 1, 6, 11, 16 (difference 5).
16 – 5 = 11; 11 – 5 = 6; 6 – 5 = 1
A line like this is called an arithmetic sequence.
Difference that has a fixed value is called different and is denoted by b. In general it can be said that if n is the formula for the term syllabus n an arithmetic sequence, then it applies
b = U _{ n } – U _{ n1 }
If the first term (U _{ 1 }) is denoted by and the difference is denoted by b ]then the formula for the term n that line can be derived as follows.
U _{ 1 } = a
U _{ 2 } = U 1 + b = a + b
U _{ 3 } = U _{ 2 } ] + b = (a + b) + b = a + 2b
[1945908] U _{ 4 } = U _{ 3 } + b = (a + 2 b) + b = a + 3b
U _{ 5 } = U _{ 4 } + b = (a + 3 b) + b = a + 4b
So, we can form the term [n9009008] n from the arithmetic sequence is
U _{ n } = a + (n – 1) b
Description:
_{ n [194590011] n }
] The first tribe
[b] [19959002] different
[n= n = many tribes
If arithmetic line terms are added together, arithmetic series will be obtained. . Suppose _{ 1 }U _{ 2 }U _{ 3 }……., U _{ n } is a the terms of the arithmetic sequence, then U _{ 1 } + U _{ 2 } + U _{ 3 } + … + U _{ n [19659034] is called arithmetic series. The arithmetic series of is written with notation S n with [1945985] U n = a + (n – 1) b. [19659002] So, the general formula for arithmetic series is }
S = 1/2 n (a + U _{ n })
S _{ n } = 1/2 n (2a + (n – 1) b)
Description:
_{ n } = number of arithmetic series n
[a] the first tribe
[b] different
U _{ n [1945900] = } n ]
n = many tribes
Determine the term [arithmetic] n arithmetic sequence if it is known the formula for the number of the first term. can be determined by the following formula.
U _{ n } = S _{ n } – S _{ n1 }

Arithmetic Line Middle Rows
If the following arithmetic sequences are known: U _{ 1 }U _{ 2 }U _{ 3 }…… ., U _{ n, } with the number of arithmetic sequence terms which are odd, then there is a term right in the middle of the row which divides the line into 2 equal parts. Suppose the middle term of the sequence is U _{ t } . So to look for the middle tribe is as follows: [19659002]
Information:
U _{ t } = the middle term of the arithmetic sequence
a = first tribe
_{ n } = the last number of arithmetic sequences with many n odd

Inserts on Arithmetic Rows
Suppose U _{ 1 }U _{ 2 }U _{ 3 }… …., U _{ n, } is an arithmetic sequence with the initial term U _{ 1. } If between two consecutive terms are inserted in such a number that a new arithmetic sequence is formed, then the difference in the new arithmetic sequence formed is as follows.
Description:
b '= new difference after insertion
b = different before insertion
k = inserted number [19659004] Geometry Formulas
Try to observe the lines 1, 2, 4, 8, 16, … See that the next term is obtained by multiplying 2 in the previous term. This line includes the geometry sequence.
So, it can be concluded in general, geometric sequence is a sequence of numbers for which each tribe is derived from the previous term multiplied by a fixed number (constant). The fixed number is called the ratio (symbol) and symbolized by r.
If U _{ 1 }U _{ 2 } , U _{ 3 }U _{ 4 } ……., U _{ n } geometric sequence with U _{ n } is the formula ] n, and the ratio r, applies:
Geometry Formulas [1945956]
The general formula for the term [n=1945900] n geometric sequence with the first term ( U _{ 1 } stated and the ratio r, can be derived as follows.
U _{ 1 } = a [19459]
U _{ 2 } = U _{ 1 } xr = ar
U _{ 3 } = U _{ 2 } xr = ar²
U _{ 4 } = U _{ 3 } xr = ar ^{ 3 }
::
U _{ n } = U _{ n1 } xr = ar ^{ n2 } [194590025] ] xr. [ar] ^{ n1 }
Thus obtained geometry sequences [a]ar, ar², …, ar ^{ n1 }.
Thus, the general formula for the term [n9009008] n geometry sequence is
U _{ n } = ar ^{ n1 }
Note:
= [1945995] tribe n
a = the first tribe
r = ratio
n = many tribes
If U _{ 1 }U _{ 2 }U _{ 3 }U _{ 4 } ……, U _{ n } is a geometric sequence so U _{ 1 + } U _{ 2 + } U _{ 3 + } U + … .. + U _{ n } is the geometric series [1945995] with U _{ U n [19659034] = n [1] . }
The general formula for determining the number of the first term of the geometric series can be derived as follows.
Suppose S _{ n } notation of the number of the first tribe.
S [194590] n = U _{ 1 + } U _{ 2 + } U _{ 3 } + … .. U _{ n }
S _{ n } = a _{ + } ar _{ + } ar ^{ 2 } _{ } + … … + ar ^{ n1 } ………………………………………………………………… (1)
If both segments are multiplied r, hence
rS = _{ + } ar [1945901] 2 2 _{ }_{ + } ar ^{ 3 } + … + ar ^{ n } ………………………………………………………… …… (2)
From the difference in equations (1) and (2), we can get
the geometric series formula [1945958]
So, the formula number n The first term of the geometric series, namely as following.
[1945995] Geometry Formulas and Geometry Series ” width=”369″ height=”175″/>
Description:
S _{ n } = the number of the first tribe [19659002] the first tribe
r = ratio
[n= n = many tribes

Middle Tribe Sequences or Series of Geometry
Suppose known geometric sequences the following: [1945909] U _{ 1 }U _{ 2 }U _{ 3 }U _{ 4 } ……., U _{ n [19659013] (the number of tribes is odd). The middle term of the sequence is U 1945. T, } the formula is as follows.
Description:
U = geometry middle line
a = first tribe
U _{ n } = the last syllable of the geometric sequence with the number n odd [199009]

Inserts on the Geometry Row
If the geometry sequence is inserted in such a number between two consecutive terms, so that a new line of geometry is formed. So in searching for the new ratio it can be formulated as follows.
Description:
r '= new ratio after insertion [19659002] r = ratio before insertion
k = [1945995] inserted numbers

Geometry Series No Infinite
An infinite geometry series is a geometry series that cannot be counted by many of its tribes. Consider the following example!
a. 1 + 2 + 4 + 8 + …
b. 5 – 10 + 20 – 40 + …
c. 1 + 1/2 + 1/4 + 1/8 + …
d. 9 – 3 + 1 – 1/3 + …
The above series are examples of infinite geometry series. Consider the examples a and b. The series is a divergent series, [9009009] which is a series that does not lead to a certain value and has a ratio r with  r  > 1.
Furthermore, examples c and d are convergent series, which is a series that leads to a certain value and has a ratio r with  r  <1.
In a convergent sequence, the number of the tribes will not exceed a certain price, but will approach a certain price. This particular price is called the infinite number of terms denoted by S _{ ∞. } The value S _{ pendekatan } is the limit value of the entire term (S 1945911) n ) with [n9009008] approaching infinity. Therefore, the infinite series formula can be derived from the geometry series with the first term n → ∞.
Because of convergent series ( r  <1), for n → ∞, then ar ^{ n } → 0 so
So, the formula for the number of infinite geometry is
Sample Questions and Discussion
1. Determine the 6th and 10th terms of the rows 3, 2, 7, 12, …
Discussion:
a = 3
] b [1945911] = U _{ n } – U _{ n1 } = 2 – (3) = 5
U _{ n } = a + (n1) b, then:
U _{ 6 } = (3) + (61) 5 = 22
U _{ 10 } = (3) + (101) 5 = 42
2. Find the first 90 terms of the series 2 + 4 + 6 + 8 + …
Discussion:
a = 2
b = 422 = 2
= 90
hence,
S _{ n } = 1/2 n (2 a + (n900] n1 b
S _{ n } = 1/2 x 90 (2 (2 ) + (90 – 1) 2)
S _{ n } = 45 (4 + 178)
S _{ n } = 45 (182) [19659002] S n = 8,190
So, the number of the first 90 terms of the series is 8,190.
3. Arithmetic sequence 3, 5, 7, 9, … 1,007.
Determine the middle tribe of the sequence.
Discussion:
a = 3
] b = 57 = 2
U n = 1,007, then:
U _{ t } = 1/2 ( a [1945990] + U _{ n [1945911]] }
U t = 1/2 (3 + 1,007)
U _{ t } = 1/2 (1,010 )
U _{ t } = 505
4. Known row 2, 12, 22, 32, … Between two consecutive terms 4 numbers are inserted such that new arithmetic sequences are formed. Determine:
a. New difference
b. The nth term formula
c. The number n first term of the new arithmetic sequence
Discussion:
a.
b. The nth term formula is
U n a + (n1) b
U _{ n } = 2 + (n1) 2
U _{ n } = 2 + (2n2)
U _{ n } = 2n
c. The number n of the first term is
S n = 1/2 n ( a + U _{ n })
S _{ n } = 1/2 n (2 + 2n)
S _{ n } = n (n + 1)
5. Find the first term, the ratio, and the 8th term from the geometric sequence 2, 6, 18, 54, ….
Discussion:
a = 2
r = U _{ 2 } / U _{ 1 } = 6/2 = 3
then,
U _{ n } = ] ar ^{ n1 }
U _{ 8 ar 81 }
U _{ 8 } = 2 ( 3 ^{ 81 }
U _{ 8 } = 2 (3 ^{ 7 })
U _{ 8 } = 2 (2,187)
U _{ 8 } = 4,374
6. Determine the number of geometry series 2 + 4 + 8 + 16 + … (8 tribes)
Discussion:
= 2
r = U _{ 2 } / U _{ ] 1 } = 4/2 = 2 ( r> 1)
The number of series up to the first 8 terms, means n = 8
So, the number of the first 8 tribes of the geometry series is 510.
7. Known geometry sequences 1/8, 1/4, 1/2, 1, 2, 4, …, 2,048. Determine the middle tribe of the sequence.
Discussion:
a = 1/8
r = U _{ 2 } / U _{ 1 } = 1/8: 1/4 = 2
U n = 2,048
then,
8. The geometric sequence 1/32, 1/16, 1/8, 1/4, … is known. Between two consecutive terms inserted 3 numbers in such a way as to form a new geometric sequence. Determine:
a. Positive ratio of new geometry sequence
b. The nth term formula for the new geometric sequence
c. The number n first term of the new geometry sequence.
Discussion: [19459900]
Geometry sequence 1/32, 1/16, 1/8, 1/4, … or it can also be presented 2 ^{ 5 }2 ^{ 4 }2 ^{ 3 }2 ^{ 2 }….
a . Notice the two consecutive terms, for example U _{ 1 } and U _{ 2 } = 1/32 and 1/16, then:
b. The nth term formula is
U = ar ^{ n1 }
U _{ n } = 2 ^{ 5 } (2 ^{ n / 41 }
U _{ n } = 2 ^{ n / 46 }
c. The number of the first tribe was
9. Determine the infinite number of terms from the geometry series 1 + 1/2 + 1/4 + 1/8 + …
Discussion:
a = 1
r = U _{ 2 } / U _{ 1 } = 1/2: 1 = 1/2
then,
10. The first term of a geometric series is 2 and the number to infinity is 4. Find the ratio.
Discussion:
a = 2
S _{ ∞ } = 4
then, we substitute it in the formula S _{ ∞ }
That is the description of arithmetic and geometric formulas along with examples of questions and discussion. Hopefully useful!
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