Geometric progression

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In mathematics, the geometric series , also known as geometric sequence , is the sequence of numbers in which each term after the first is found by multiplying the previous number with the fixed instead of the so-called zero general ratio . For example, the order 2, 6, 18, 54,... is a geometric series with the general ratio 3. Similarly 10, 5, 2.5, 1.25,... is a geometric series with a general ratio of 1/2.

Contoh dari deret geometrik adalah kekuatan r ^k dari angka tetap r , seperti 2 ^{< i> k} dan 3 ^k . Bentuk umum dari urutan geometrik adalah

${\ displaystyle a, \ ar, \ ar ^ {2}, \ ar ^ {3}, \ ar ^ {4}, \ \ ldots}  ÂÂ$

where is r ? 0 is the common ratio and a is the scale factor, equal to the initial value of the order.

Video Geometric progression

Basic properties

The n -th istilah dari urutan geometrik dengan nilai awal a dan rasio umum r diberikan oleh

${\ displaystyle a_ {n} = a \, r ^ {n-1}.}  ÂÂ$

Urutan geometrik seperti itu juga mengikuti relasi rekursif

${\ displaystyle a_ {n} = r \, a_ {n-1}}  ÂÂ$ untuk setiap bilangan bulat ${\ displaystyle n \ geq 1.}  ÂÂ$

Generally, to check whether the given sequence is geometric, one simply checks whether the entries in a row in the all order have the same ratios.

The general ratio of the geometric sequence may be negative, resulting in alternating sequences, with numbers switching from positive to negative and back. Example

1, -3, 9, -27, 81, -243,...

is a geometric series with a general ratio of -3.

The behavior of the geometric sequence depends on the value of the general ratio If the general ratio is:

• Positively, all conditions will be the same sign as the original term.
• Negative, the term will alternate between positive and negative.
• Greater than 1, there will be exponential growth leading to infinite positive or negative (depending on the sign of the original term).
• 1, its development is a constant sequence.
• Between -1 and 1 but not zero, there will be exponential decay to zero.
• -1, the development is alternating sequence
• Less than -1, for the absolute value there is an exponential growth against the unbound (unsigned), because of the alternate mark.

Geometric sequence (with a general ratio not equal to -1, 1 or 0) shows exponential growth or exponential decay, as opposed to linear growth (or decrease) of arithmetic series such as 4, 15, 26, 37, 48,... (with common difference 11). This result was taken by T.R. Malthus as the mathematical basis of his Principle of Population . Note that two types of related developments: exponentiating each term from an arithmetic progression produces a geometric progression, while taking the logarithm of each term in geometric progression with a positive general ratio resulting in an arithmetic progression.

Hasil yang menarik dari definisi pengembangan geometrik adalah bahwa untuk setiap nilai rasio umum, tiga istilah berturut-turut a , b dan c akan memenuhi persamaan berikut:

${\ displaystyle b ^ {2} = ac}  ÂÂ$

where b is considered a geometric mean between a and c .

Maps Geometric progression

Geometric series

Sebuah deret geometrik adalah penjumlahan angka-angka dalam deret geometrik. Sebagai contoh:

${\ displaystyle 2 10 50 250 = 2 2 \ kali 5 2 \ kali 5 ^ {2} 2 \ kali 5 ^ {3}.}  ÂÂ$

Membiarkan a menjadi istilah pertama (di sini 2), n menjadi jumlah istilah (di sini 4), dan r menjadi konstanta bahwa setiap istilah dikalikan dengan untuk mendapatkan istilah berikutnya (di sini 5), penjumlahan diberikan oleh:

${\ displaystyle {\ frac {a (1-r ^ {n})} {1-r}}}  ÂÂ$

Dalam contoh di atas, ini memberi:

${\ displaystyle 2 10 50 250 = {\ frac {2 (1-5 ^ {4})} {1-5}} = {\ frac {- 1248} {- 4}} = 312.}  ÂÂ$

Rumus bekerja untuk bilangan real a dan r (kecuali r = 1, yang menghasilkan pembagian dengan nol). Sebagai contoh:

${\ displaystyle -2 \ pi 4 \ pi ^ {2} -8 \ pi ^ {3} = - 2 \ pi (- 2 \ pi) ^ {2 } (- 2 \ pi) ^ {3} = {\ frac {-2 \ pi (1 - (- 2 \ pi) ^ {3})} {1 - (- 2 \ pi)}} = {\ frac {-2 \ pi (1 8 \ pi ^ {3})} {1 2 \ pi}} \ approx -214.855.}  ÂÂ$

Since the derivation (below) does not depend on the real a and r , it applies to complex numbers as well.

Derivation

Untuk memperoleh rumus ini, pertama-tama tuliskan deret geometrik umum sebagai:

${\ displaystyle \ sum _ {k = 1} ^ {n} ar ^ {k-1} = ar ^ {0} ar ^ {1} ar ^ { 2} ar ^ {3} \ cdots ar ^ {n-1}.}  ÂÂ$

karena semua ketentuan lainnya dibatalkan. Jika r ? 1, kita dapat mengatur ulang di atas untuk mendapatkan rumus yang mudah digunakan untuk deret geometrik yang menghitung jumlah n istilah:

${\ displaystyle \ sum _ {k = 1} ^ {n} ar ^ {k-1} = {\ frac {a (1-r ^ {n})} {1-r}}.}  ÂÂ$

Rumus terkait

Jika seseorang memulai jumlah tidak dari k = 1, tetapi dari nilai yang berbeda, katakanlah m , maka

${\ displaystyle \ sum _ {k = m} ^ {n} ar ^ {k} = {\ frac {a (r ^ {m} -r ^ {n 1})} {1-r}}.}  ÂÂ$

Membedakan rumus ini dengan hormat kepada r memungkinkan kita untuk sampai pada rumus untuk jumlah formulir

${\ displaystyle G_ {s} (n, r): = \ jumlah _ {k = 0} ^ {n} k ^ {s} r ^ {k}.}  ÂÂ$

Sebagai contoh:

${\ displaystyle {\ frac {d} {dr}} \ jumlah _ {k = 0} ^ {n} r ^ {k} = \ jumlah _ {k = 1 } ^ {n} kr ^ {k-1} = {\ frac {1-r ^ {n 1}} {(1-r) ^ {2}}} - {\ frac {(n 1) r ^ {n}} {1-r}}.}  ÂÂ$

Untuk deret geometrik yang hanya memuat kekuatan r dikalikan dengan  1 - r ² Â:

${\ displaystyle (1-r ^ {2}) \ jumlah _ {k = 0} ^ {n} ar ^ {2k} = a-ar ^ {2n 2 }.}  ÂÂ$

Kemudian

${\ displaystyle \ sum _ {k = 0} ^ {n} ar ^ {2k} = {\ frac {a (1-r ^ {2n 2})} {1-r ^ {2}}}.}  ÂÂ$

Equivalently, take Ã, r ² as the general ratio and use the standard formulation.

Untuk seri dengan kekuatan ganjil r

${\ displaystyle (1-r ^ {2}) \ jumlah _ {k = 0} ^ {n} ar ^ {2k 1} = ar-ar ^ {2n 3}}  ÂÂ$

dan

${\ displaystyle \ sum _ {k = 0} ^ {n} ar ^ {2k 1} = {\ frac {ar (1-r ^ {2n 2} )} {1-r ^ {2}}}.}  ÂÂ$

Rumus yang tepat untuk penjumlahan umum ${\ displaystyle G_ {s} (n, r)}  ÂÂ$ ketika ${\ displaystyle s \ in \ mathbb {N}}  ÂÂ$ diperluas oleh nomor Stirling dari jenis kedua sebagai

$            {\displaystyle         G_{            s          }        (        n        ,        r        )        =        \underset{            j            =            0          }{\overset{            s          }{?}}                  \left\{                      \genfrac{}{}{0ex}{}{s}{j}                    \right\}                x^{            j          }                  \frac{d^{                j        }}{}}$

Source of the article : Wikipedia