This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Hapa kuna hatua za kutatua maswali: 1. The n^th term of a geometric progression is given by the formula 18((4)/(3))^n. Determine the sum of the first four terms of the progression. Step 1: Tafuta neno la kwanza (a) na uwiano wa kawaida (r). Neno la n^th limetolewa kama T_n = 18((4)/(3))^n. Kwa n=1, neno la kwanza ni: T_1 = 18((4)/(3))^1 = 18 × (4)/(3) = 6 × 4 = 24 Kwa n=2, neno la pili ni: T_2 = 18((4)/(3))^2 = 18 × (16)/(9) = 2 × 16 = 32 Uwiano wa kawaida r ni (T_2)/(T_1): r = (32)/(24) = (4)/(3) Hivyo, a = 24 na r = (4)/(3). Step 2: Tumia fomula ya jumla ya maneno ya kwanza n ya mfuatano wa kijiometri. Fomula ni S_n = (a(r^n - 1))/(r - 1) kwa r > 1. Tunataka jumla ya maneno manne ya kwanza, hivyo n=4. S_4 = (24((4)/(3))^4 - 1)(4)/(3) - 1 S_4 = (24(256)/(81) - 1)(1)/(3) S_4 = (24(256 - 81)/(81))(1)/(3) S_4 = (24(175)/(81))(1)/(3) S_4 = 24 × (175)/(81) × 3 S_4 = 24 × (175)/(27) S_4 = (8 × 175)/(9) S_4 = (1400)/(9) (1400)/(9) 2. Without using mathematical tables or calculator evaluate (_2 256 - _5 625)/(_3 27 + _2 128) Step 1: Rahisisha kila neno la logariti katika nambari. _2 256 = _2 2^8 = 8 _5 625 = _5 5^4 = 4 Step 2: Rahisisha kila neno la logariti katika denomineta. _3 27 = _3 3^3 = 3 _2 128 = _2 2^7 = 7 Step 3: Badilisha thamani zilizorahisishwa kwenye usemi na utathmini. (_2 256 - _5 625)/(_3 27 + _2 128) = (8 - 4)/(3 + 7) = (4)/(10) = (2)/(5) (2)/(5) 3. Express 510^ in surd form hence simplify sqrt(5)1- 510^ leaving your answer in the form asqrt(b)+csqrt(d) where a, b, c and d are integers. Step 1: Eleza 510^ katika surd form. Tunajua kuwa ( + 360^) = . 510^ = (510^ - 360^) = 150^ Tunajua pia kuwa (180^ - ) = -. 150^ = (180^ - 30^) = - 30^ Thamani ya 30^ ni sqrt(3)2. Hivyo, 510^ = -sqrt(3)2 Step 2: Badilisha thamani ya 510^ kwenye usemi sqrt(5)1- 510^. sqrt(5)1 - (-sqrt(3)2) = sqrt(5)1 + sqrt(3)2 Ili kurahisisha denomineta, pata kiashiria cha kawaida: sqrt(5)(2)/(2) + sqrt(3)2 = sqrt(5)2+sqrt(3)2 Geuza na zidisha: = 2sqrt(5)2+sqrt(3) Step 3: Rationalize denomineta na urahisishe hadi fomu asqrt(b)+csqrt(d). Zidisha nambari na denomineta kwa kiambishi cha denomineta, ambacho ni 2-sqrt(3). 2sqrt(5)2+sqrt(3) × 2-sqrt(3)2-sqrt(3) = 2sqrt(5)(2-sqrt(3))(2+sqrt(3))(2-sqrt(3)) Panua nambari: 2sqrt(5)(2-sqrt(3)) = 4sqrt(5) - 2sqrt(5)sqrt(3) = 4sqrt(5) - 2sqrt(15) Panua denomineta (tumia tofauti ya miraba, (x+y)(x-y) = x^2 - y^2): (2+sqrt(3))(2-sqrt(3)) = 2^2 - (sqrt(3))^2 = 4 - 3 = 1 Hivyo, usemi uliorahisishwa ni: 4sqrt(5) - 2sqrt(15)1 = 4sqrt(5) - 2sqrt(15) Hii iko katika fomu asqrt(b)+csqrt(d) ambapo a=4, b=5, c=-2, na d=15. 4sqrt(5) - 2sqrt(15) That's 2 down. 3 left today — send the next one.