Tambayoyi Misali Game da Jerin Abubuwa da Jerin Abubuwa
Jerin kalmomi da jerin kalmomi muhimman ra'ayoyi ne a fannin lissafi, waɗanda ake yawan samu daga makarantar firamare har zuwa kwaleji. Jerin kalmomi tarin lambobi ne da aka tsara bisa ga takamaiman ƙa'ida, yayin da jeri shine jimlar kalmomin wannan jerin. A cikin wannan labarin, za mu tattauna misalai da yawa na matsaloli kuma mu tattauna jerin kalmomi da jerin kalmomi.
Misali na 1: Jerin Lissafi
Tambaya:
An ba da jerin lissafi tare da kalma ta farko (a) = 3 da bambanci (d) = 5. A tantance:
1. Kalma ta 10 ta jerin.
2. Jimlar kalmomin farko 20 na jerin.
Tattaunawa:
1. Zagaye na 10
Tsarin da ake amfani da shi don zangon n na jerin lissafi shine:
\[
U_n = a + (n-1)d
\]
Don zangon karatu na 10 (U_10):
\[
U_{10} = 3 + (10-1) \cdot 5 = 3 + 45 = 48
\]
2. Jimlar Kalmomi 20 na Farko
Tsarin jimlar kalmomin n na farko (S_n) na jerin lissafi shine:
\[
S_n = \frac{n}{2} (2a + (n-1)d)
\]
Jimlar kalmomin farko 20 (S_20):
\[
S_{20} = \frac{20}{2} (2 \cdot 3 + (20-1) \cdot 5) = 10 (6 + 95) = 10 \cdot 101 = 1010
\]
Misali na 2: Jerin Lissafi
Tambaya:
An ba da jerin siffofi na lissafi tare da kalma ta farko (a) = 4 da rabon (r) = 2. A ƙayyade:
1. Kalma ta 6 ta jerin.
2. Jimlar kalmomin farko 8 na jerin.
Tattaunawa:
1. Zagaye na 6
Tsarin da za a yi amfani da shi don zangon n na jerin siffofi shine:
\[
U_n = a \cdot r^{(n-1)}
\]
Don zangon karatu na 6 (U_6):
\[
U_{6} = 4 \cdot 2^{(6-1)} = 4 \cdot 2^5 = 4 \cdot 32 = 128
\]
2. Jimlar Kalmomi 8 na Farko
Tsarin jimlar kalmomin n na farko (S_n) na jerin siffofi na lissafi shine:
\[
S_n = a \frac{r^n – 1}{r – 1}
\]
Jimlar kalmomin farko 8 (S_8):
\[
S_{8} = 4 \frac{2^8 – 1}{2 – 1} = 4 \frac{256 – 1}{1} = 4 \cdot 255 = 1020
\]
Misali na 3: Jerin Convergent Geometric Infinite
Tambaya:
An ba da jerin siffofi na lissafi tare da kalma ta farko (a) = 1 da rabon (r) = 1/2. A ƙayyade jimlar jerin marasa iyaka.
Tattaunawa:
Tsarin jimlar jerin marasa iyaka (S_∞) na jerin geometric masu haɗuwa shine:
\[
S_{\infty} = \frac{a}{1 – r}
\]
Don haka ga wannan jerin:
\[
S_{\infty} = \frac{1}{1 – \frac{1}{2}} = \frac{1}{\frac{1}{2}} = 2
\]
Misali na 4: Jerin Lambobin Murabba'i da Jerin Lambobin Murabba'i
Tambaya:
An ba da jerin lambobi murabba'i tare da kalma ta farko (U_1) = 1, kalma ta biyu (U_2) = 4, da kalma ta uku (U_3) = 9. Kayyade kalma ta 5 ta jerin. Shin wannan jerin lissafi ne ko jerin lissafi? Bayyana.
Tattaunawa:
1. Zagaye na 5
Tsarin jerin lambobi murabba'i shine:
\[
U_n = n^2
\]
Don zangon karatu na 5 (U_5):
\[
U_{5} = 5^2 = 25
\]
2. Nau'in Layi
Domin duba ko wannan jerin yana da lissafi ko na lissafi, duba bambanci tsakanin kalmomin (bambancin gama gari) da rabo tsakanin kalmomin:
– Bambanci tsakanin kalmomi (d):
\[
U_2 – U_1 = 4 – 1 = 3 \\
U_3 – U_2 = 9 – 4 = 5
\]
Tunda bambancin ba ya canzawa, wannan jerin ba lissafi bane.
– Rabon kabilu (r):
\[
\frac{U_2}{U_1} = \frac{4}{1} = 4 \\
\frac{U_3}{U_2} = \frac{9}{4} = 2.25
\]
Tunda rabon da ke tsakanin kalmomi ba shi da tabbas, wannan jerin ba shi da tsari na lissafi.
Don haka, wannan jerin lambobin murabba'i ba lissafi ba ne ko kuma jerin siffofi, a'a, tsari ne na musamman wanda ke bin tsarin lambobin murabba'i.
Misali na 5: Jerin Lissafi Mara iyaka
Tambaya:
Shin zai yiwu a ƙididdige jimlar jerin lissafi marasa iyaka? Idan haka ne, bayar da misali. Idan ba haka ba, bayyana dalilin.
Tattaunawa:
Ba kamar jerin lissafi ba, jerin lissafi marasa iyaka yawanci ba su da jimlar iyaka. Wannan saboda kowace kalma tana ƙaruwa ko raguwa a layi, don haka jimlar tana ci gaba da girma ba tare da wani lokaci ba.
Misali, yi la'akari da jerin lissafi marasa iyaka tare da kalma ta farko 1 da bambanci na gama gari 1:
\[
1 + 2 + 3 + 4 + \ldot
\]
Idan muka yi ƙoƙarin taƙaita su, zai bayyana cewa jerin ba za su haɗu zuwa wani ƙima mai ƙayyadadden ƙima ba, amma za su kusanci rashin iyaka. Saboda haka, jimlar jerin lissafi mara iyaka, gabaɗaya, ba ta da iyaka kuma ba za a iya ƙididdige ta kamar jerin lissafi masu haɗuwa ba.
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A cikin wannan labarin, mun tattauna misalai da dama kuma mun tattauna jerin da jerin. Mun yi nazari kan jerin lissafi da na lissafi, mun ga yadda ake ƙididdige jimlar kalmomin farko da jimlar kalmomin farko, kuma mun amsa tambayoyi game da jerin marasa iyaka. Ta hanyar fahimtar waɗannan mahimman ra'ayoyi da misalai, muna fatan za ku ji ƙarin kwarin gwiwa yayin da kuke kusantar jerin da jerin a cikin lissafi.
