Berikut ini adalah pertanyaan dari callyzta1990 pada mata pelajaran Matematika untuk jenjang Sekolah Menengah Atas
Jawaban dan Penjelasan
Berikut ini adalah pilihan jawaban terbaik dari pertanyaan diatas.
(i)
U1 + U2 + U3 + U4 = 30
a + ar + ar² + ar³ = 30
a(1 + r + r² + r³) = 30
a(r + 1)(r² + 1) = 30 (pers.1)
(ii)
U1² + U2² + U3² + U4² = 304
a² + a²r² + a²r⁴ + a²r⁶ = 304
a²(1 + r² + r⁴ + r⁶) = 304
a²(r² + 1)(r⁴ + 1) = 304 (pers.2)
(pers.1)
a(r + 1)(r² + 1) = 30
(a(r + 1)(r² + 1))² = 30²
a²(r + 1)²(r² + 1)² = 900 (pers.3)
(pers.3 / pers.2)
[a²(r + 1)²(r² + 1)²]/[a²(r² + 1)(r⁴ + 1) = 900/304
[(r + 1)²(r² + 1)]/(r⁴ + 1) = 45/17
17(r + 1)²(r² + 1) = 45(r⁴ + 1)
17(r² + 2r + 1)(r² + 1) = 45r⁴ + 45
17(r⁴ + r² + 2r³ + 2r + r² + 1) = 45r⁴ + 45
17(r⁴ + 2r³ + 2r² + 2r + 1) = 45r⁴ + 45
17r⁴ + 34r³ + 34r² + 34r + 17 = 45r⁴ + 45
45r⁴ + 45 - 17r⁴ - 34r³ - 34r² - 34r - 17 = 0
28r⁴ - 34r³ - 34r² - 34r + 28 = 0
2(14r⁴ - 17r³ - 17r² - 17r + 14) = 0
2(r - 2)(r - 1/2)(14r² + 18r + 14) = 0
r - 2 = 0
r = 2
atau
r - 1/2 = 0
r = 1/2
dan
14r² + 18r + 14 = 0 (tidak termasuk karena D < 0
D = 18² - 4(14)(14) = -460
r = 1/2
a(r + 1)(r² + 1) = 30
a(1/2 + 1)((1/2)² + 1) = 30
a(1/2 + 2/2)(1/4 + 4/4) = 30
a(3/2)(5/4) = 30
a(15/8) = 30
a = 30 × 8/15
a = 16
U1 = 16
U2 = 8
U3 = 4
U4 = 2
U1³ + U2³ + U3³ + U4³
= 16³ + 8³ + 4³ + 2³
= 4096 + 512 + 64 + 8
= 4680
r = 2
a(r + 1)(r² + 1) = 30
a(2 + 1)(2² + 1) = 30
a(3)(4 + 1) = 30
a(3)(5) = 30
15a = 30
a = 30/15
a = 2
U1 = 2
U2 = 4
U3 = 8
U4 = 16
U1³ + U2³ + U3³ + U4³
= 2³ + 4³ + 8³ + 16³
= 8 + 64 + 512 + 4096
= 4680
Jadi jawabannya adalah C
![(i) U1 + U2 + U3 + U4 = 30a + ar + ar² + ar³ = 30a(1 + r + r² + r³) = 30a(r + 1)(r² + 1) = 30 (pers.1)(ii)U1² + U2² + U3² + U4² = 304a² + a²r² + a²r⁴ + a²r⁶ = 304a²(1 + r² + r⁴ + r⁶) = 304a²(r² + 1)(r⁴ + 1) = 304 (pers.2)(pers.1)a(r + 1)(r² + 1) = 30(a(r + 1)(r² + 1))² = 30²a²(r + 1)²(r² + 1)² = 900 (pers.3)(pers.3 / pers.2)[a²(r + 1)²(r² + 1)²]/[a²(r² + 1)(r⁴ + 1) = 900/304[(r + 1)²(r² + 1)]/(r⁴ + 1) = 45/1717(r + 1)²(r² + 1) = 45(r⁴ + 1)17(r² + 2r + 1)(r² + 1) = 45r⁴ + 4517(r⁴ + r² + 2r³ + 2r + r² + 1) = 45r⁴ + 4517(r⁴ + 2r³ + 2r² + 2r + 1) = 45r⁴ + 4517r⁴ + 34r³ + 34r² + 34r + 17 = 45r⁴ + 4545r⁴ + 45 - 17r⁴ - 34r³ - 34r² - 34r - 17 = 028r⁴ - 34r³ - 34r² - 34r + 28 = 02(14r⁴ - 17r³ - 17r² - 17r + 14) = 02(r - 2)(r - 1/2)(14r² + 18r + 14) = 0r - 2 = 0r = 2atau r - 1/2 = 0r = 1/2 dan14r² + 18r + 14 = 0 (tidak termasuk karena D < 0D = 18² - 4(14)(14) = -460r = 1/2a(r + 1)(r² + 1) = 30a(1/2 + 1)((1/2)² + 1) = 30a(1/2 + 2/2)(1/4 + 4/4) = 30a(3/2)(5/4) = 30a(15/8) = 30a = 30 × 8/15a = 16U1 = 16U2 = 8U3 = 4U4 = 2U1³ + U2³ + U3³ + U4³= 16³ + 8³ + 4³ + 2³= 4096 + 512 + 64 + 8= 4680r = 2a(r + 1)(r² + 1) = 30a(2 + 1)(2² + 1) = 30a(3)(4 + 1) = 30a(3)(5) = 3015a = 30a = 30/15a = 2U1 = 2U2 = 4U3 = 8U4 = 16U1³ + U2³ + U3³ + U4³= 2³ + 4³ + 8³ + 16³= 8 + 64 + 512 + 4096= 4680Jadi jawabannya adalah C](https://id-static.z-dn.net/files/d58/d0751e229aa8b8226974e68231b20246.jpg)
![(i) U1 + U2 + U3 + U4 = 30a + ar + ar² + ar³ = 30a(1 + r + r² + r³) = 30a(r + 1)(r² + 1) = 30 (pers.1)(ii)U1² + U2² + U3² + U4² = 304a² + a²r² + a²r⁴ + a²r⁶ = 304a²(1 + r² + r⁴ + r⁶) = 304a²(r² + 1)(r⁴ + 1) = 304 (pers.2)(pers.1)a(r + 1)(r² + 1) = 30(a(r + 1)(r² + 1))² = 30²a²(r + 1)²(r² + 1)² = 900 (pers.3)(pers.3 / pers.2)[a²(r + 1)²(r² + 1)²]/[a²(r² + 1)(r⁴ + 1) = 900/304[(r + 1)²(r² + 1)]/(r⁴ + 1) = 45/1717(r + 1)²(r² + 1) = 45(r⁴ + 1)17(r² + 2r + 1)(r² + 1) = 45r⁴ + 4517(r⁴ + r² + 2r³ + 2r + r² + 1) = 45r⁴ + 4517(r⁴ + 2r³ + 2r² + 2r + 1) = 45r⁴ + 4517r⁴ + 34r³ + 34r² + 34r + 17 = 45r⁴ + 4545r⁴ + 45 - 17r⁴ - 34r³ - 34r² - 34r - 17 = 028r⁴ - 34r³ - 34r² - 34r + 28 = 02(14r⁴ - 17r³ - 17r² - 17r + 14) = 02(r - 2)(r - 1/2)(14r² + 18r + 14) = 0r - 2 = 0r = 2atau r - 1/2 = 0r = 1/2 dan14r² + 18r + 14 = 0 (tidak termasuk karena D < 0D = 18² - 4(14)(14) = -460r = 1/2a(r + 1)(r² + 1) = 30a(1/2 + 1)((1/2)² + 1) = 30a(1/2 + 2/2)(1/4 + 4/4) = 30a(3/2)(5/4) = 30a(15/8) = 30a = 30 × 8/15a = 16U1 = 16U2 = 8U3 = 4U4 = 2U1³ + U2³ + U3³ + U4³= 16³ + 8³ + 4³ + 2³= 4096 + 512 + 64 + 8= 4680r = 2a(r + 1)(r² + 1) = 30a(2 + 1)(2² + 1) = 30a(3)(4 + 1) = 30a(3)(5) = 3015a = 30a = 30/15a = 2U1 = 2U2 = 4U3 = 8U4 = 16U1³ + U2³ + U3³ + U4³= 2³ + 4³ + 8³ + 16³= 8 + 64 + 512 + 4096= 4680Jadi jawabannya adalah C](https://id-static.z-dn.net/files/d8a/255894a40e555ae03ccb0908d493803e.jpg)
![(i) U1 + U2 + U3 + U4 = 30a + ar + ar² + ar³ = 30a(1 + r + r² + r³) = 30a(r + 1)(r² + 1) = 30 (pers.1)(ii)U1² + U2² + U3² + U4² = 304a² + a²r² + a²r⁴ + a²r⁶ = 304a²(1 + r² + r⁴ + r⁶) = 304a²(r² + 1)(r⁴ + 1) = 304 (pers.2)(pers.1)a(r + 1)(r² + 1) = 30(a(r + 1)(r² + 1))² = 30²a²(r + 1)²(r² + 1)² = 900 (pers.3)(pers.3 / pers.2)[a²(r + 1)²(r² + 1)²]/[a²(r² + 1)(r⁴ + 1) = 900/304[(r + 1)²(r² + 1)]/(r⁴ + 1) = 45/1717(r + 1)²(r² + 1) = 45(r⁴ + 1)17(r² + 2r + 1)(r² + 1) = 45r⁴ + 4517(r⁴ + r² + 2r³ + 2r + r² + 1) = 45r⁴ + 4517(r⁴ + 2r³ + 2r² + 2r + 1) = 45r⁴ + 4517r⁴ + 34r³ + 34r² + 34r + 17 = 45r⁴ + 4545r⁴ + 45 - 17r⁴ - 34r³ - 34r² - 34r - 17 = 028r⁴ - 34r³ - 34r² - 34r + 28 = 02(14r⁴ - 17r³ - 17r² - 17r + 14) = 02(r - 2)(r - 1/2)(14r² + 18r + 14) = 0r - 2 = 0r = 2atau r - 1/2 = 0r = 1/2 dan14r² + 18r + 14 = 0 (tidak termasuk karena D < 0D = 18² - 4(14)(14) = -460r = 1/2a(r + 1)(r² + 1) = 30a(1/2 + 1)((1/2)² + 1) = 30a(1/2 + 2/2)(1/4 + 4/4) = 30a(3/2)(5/4) = 30a(15/8) = 30a = 30 × 8/15a = 16U1 = 16U2 = 8U3 = 4U4 = 2U1³ + U2³ + U3³ + U4³= 16³ + 8³ + 4³ + 2³= 4096 + 512 + 64 + 8= 4680r = 2a(r + 1)(r² + 1) = 30a(2 + 1)(2² + 1) = 30a(3)(4 + 1) = 30a(3)(5) = 3015a = 30a = 30/15a = 2U1 = 2U2 = 4U3 = 8U4 = 16U1³ + U2³ + U3³ + U4³= 2³ + 4³ + 8³ + 16³= 8 + 64 + 512 + 4096= 4680Jadi jawabannya adalah C](https://id-static.z-dn.net/files/d3b/ba93ff66de690b3a35249fc7bbcf9d14.jpg)
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Last Update: Thu, 20 Jul 23