Tambayoyi Misali Game da Jerin Lissafi
Jerin lissafi muhimmin ra'ayi ne a fannin lissafi wanda ke bayyana akai-akai a cikin matsaloli daban-daban, duka a makarantun sakandare da manyan makarantu. Wannan ra'ayi ya ƙunshi jerin lambobi inda kowane kalma sakamakon ƙara ko cire lamba mai ɗorewa daga kalmar da ta gabata ne. A cikin wannan labarin, za mu tattauna misalai da yawa na matsaloli da mafita don ƙarin fahimtar manufar jerin lissafi.
Fahimtar Jerin Lissafi
Jerin lissafi jeri ne wanda ke da bambanci (bambanci) akai-akai tsakanin kalmomi biyu a jere. Misali, idan jerin lissafi yana da kalma ta farko \(a\) da bambanci \(d\), to ana iya rubuta kalmomin kamar haka:
\[a, a + d, a + 2d, a + 3d, \ldots \]
Idan muna son nemo zangon n na wannan jerin, dabarar zangon n (\(U_n\)) ita ce:
\[ U_n = a + (n-1)d \]
A halin yanzu, ana iya ƙididdige jimlar kalmomin n na farko na jerin lissafi (\(S_n\)) ta amfani da dabarar:
\[ S_n = \frac{n}{2} (2a + (n-1)d) \]
Tambayoyi da Tattaunawa Samfura
Misali Tambaya ta 1
Tambaya: An ba da jerin lissafi mai taken farko \(a = 5\) da kuma bambancin da aka saba \(d = 3\). Nemo jumla ta 10 ta jerin.
Tattaunawa:
Domin nemo kalma ta 10 (\(U_{10}\)), za mu iya amfani da dabarar kalma ta n:
\[ U_{10} = a + (10-1)d \]
\[ U_{10} = 5 + (9 \cdot 3) \]
\[ U_{10} = 5 + 27 \]
\[ U_{10} = 32 \]
Don haka, zango na 10 na jerin shine 32.
Misali Tambaya ta 2
Tambaya: Nemo jimlar kalmomi 15 na farko na jerin lissafi waɗanda kalmar farko ita ce \(a = 4\) kuma bambancin da aka saba shine \(d = 7\).
Tattaunawa:
Domin samun jimlar kalmomi 15 na farko (\(S_{15}\)), za mu iya amfani da dabarar don jimlar kalmomi na n na farko:
\[ S_{15} = \frac{15}{2} (2a + (15-1)d) \]
\[ S_{15} = \frac{15}{2} (2 \cdot 4 + 14 \cdot 7) \]
\[ S_{15} = \frac{15}{2} (8 + 98) \]
\[ S_{15} = \frac{15}{2} \cdot 106 \]
\[ S_{15} = 15 \cdot 53 \]
\[ S_{15} = 795 \]
Don haka, jimillar sharuɗɗan farko 15 na jerin shine 795.
Misali Tambaya ta 3
Tambaya: An san cewa zango na 5 na jerin lissafi shine 20 kuma zango na 12 shine 48. Nemo zango na farko (\(a\)) da bambancin da aka saba gani (\(d\)) na jerin.
Tattaunawa:
Daga sharuɗɗan da aka bayar:
\[ U_5 = a + 4d = 20 \]
\[ U_{12} = a + 11d = 48 \]
Muna da daidaiton layi guda biyu tare da masu canji guda biyu waɗanda za mu iya warwarewa:
1. \( a + 4d = 20 \)
2. \( a + 11d = 48 \)
Daga lissafi na 1, za mu iya bayyana \(a\) ta hanyar \(d\):
\[ a = 20 – 4d \]
Yanzu mun maye gurbin \(a\) zuwa lissafi na 2:
\[ 20 – 4d + 11d = 48 \]
\[ 20 + 7d = 48 \]
\[ 7d = 28 \]
\[ d = 4 \]
Yanzu mun maye gurbin ƙimar \(d\) zuwa lissafin \(a = 20 – 4d\):
\[ a = 20 – 4 \cdot 4 \]
\[ a = 20 – 16 \]
\[ a = 4 \]
Don haka, zangon farko na jerin shine 4 kuma bambancin da aka saba dashi shine 4.
Misali Tambaya ta 4
Tambaya: Kalmomi nawa ake buƙata don jerin lissafi tare da kalma ta farko \(a = 2\) da bambanci na gama gari \(d = 5\) don a jimlace zuwa 200?
Tattaunawa:
A wannan yanayin, muna buƙatar nemo jimlar kalmomin n na farko (\(S_n\)) wanda yayi daidai da 200. Yi amfani da dabarar don jimlar kalmomin n na farko:
\[ S_n = \frac{n}{2} (2a + (n-1)d) = 200 \]
Maye gurbin ƙimar \(a\) da \(d\):
\[ \frac{n}{2} (2 \cdot 2 + (n-1) \cdot 5) = 200 \]
\[ \ frac {n}{2} (4 + 5n - 5) = 200 \]
\[ \frac{n}{2} (5n - 1) = 200 \]
\[ n (5n – 1) = 400 \]
Wannan lissafi ne na kwata-kwata. Domin magance shi, muna canza siffarsa:
\[ 5n^2 – n – 400 = 0 \]
Yi amfani da dabarar quadratic \(ax^2 + bx + c = 0\):
\[ n = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \]
A wannan yanayin \(a = 5\), \(b = -1\), da \(c = -400\):
\[ n = \frac{-(-1) \pm \sqrt{(-1)^2 – 4 \cdot 5 \cdot (-400)}}{2 \cdot 5} \]
\[ n = \frac{1 \pm \sqrt{1 + 8000}}{10} \]
\[ n = \frac{1 \pm \sqrt{8001}}{10} \]
Darajar \(\sqrt{8001}\) ta kusa da 89.42, to:
\[ n = \frac{1 \pm 89.42}{10} \]
Muna ɗaukar kyawawan dabi'u:
\[n = \ frac{1 + 89.42}{10} \]
\[ n \approx \frac{90.42}{10} \]
\[ n \kimanin 9.042 \]
Don haka, adadin kalmomin da ake buƙata shine kalmomi 9 (idan an zagaye su).
Kammalawa
Jerin lissafi muhimmin batu ne a fannin lissafi. Fahimtar kalma ta farko, bambancin da aka saba gani, kalma ta n, da jimlar kalmomin n na farko yana da matuƙar amfani wajen magance matsaloli iri-iri. Ta hanyar amfani da misalai da tattaunawa da ke sama, ana fatan masu karatu za su fahimci muhimman ra'ayoyi na jerin lissafi.
