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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Haya, hebu tutatue maswali hayo. Jibu la Swali la 27: Ujazo wa frustum ni: 46629.52 cm^3 --- Swali la 28: Jumla ya n-vigezo vya kwanza vya A.P. inatolewa na S_n = 2n^2 - n. Tafuta kigezo cha kwanza na tofauti ya kawaida. Step 1: Tafuta kigezo cha kwanza (a_1). Kigezo cha kwanza ni jumla ya kigezo kimoja cha kwanza (S_1). S_1 = 2(1)^2 - 1 S_1 = 2(1) - 1 S_1 = 2 - 1 S_1 = 1 Hivyo, kigezo cha kwanza a_1 = 1. Step 2: Tafuta kigezo cha pili (a_2). Jumla ya vigezo viwili vya kwanza ni S_2 = a_1 + a_2. S_2 = 2(2)^2 - 2 S_2 = 2(4) - 2 S_2 = 8 - 2 S_2 = 6 Kwa kuwa S_2 = a_1 + a_2, tunapata: 6 = 1 + a_2 a_2 = 6 - 1 a_2 = 5 Step 3: Tafuta tofauti ya kawaida (d). Tofauti ya kawaida ni d = a_2 - a_1. d = 5 - 1 d = 4 Kigezo cha kwanza ni 1 na tofauti ya kawaida ni 4. --- Swali la 30: Kigezo cha 3 cha G.P. ni 18 na kigezo cha 6 ni 486. Tafuta kigezo cha kwanza na uwiano wa kawaida. Fomula ya kigezo cha n cha G.P. ni a_n = ar^n-1. Step 1: Andika milinganyo kutokana na taarifa uliyopewa. Kigezo cha 3: a_3 = ar^3-1 = ar^2 = 18 (Mlinganyo 1) Kigezo cha 6: a_6 = ar^6-1 = ar^5 = 486 (Mlinganyo 2) Step 2: Tatua kwa uwiano wa kawaida (r). Gawanya Mlinganyo 2 kwa Mlinganyo 1: (ar^5)/(ar^2) = (486)/(18) r^5-2 = 27 r^3 = 27 r = [3]27 r = 3 Step 3: Tatua kwa kigezo cha kwanza (a). Badilisha r=3 katika Mlinganyo 1: a(3)^2 = 18 9a = 18 a = (18)/(9) a = 2 Kigezo cha kwanza ni 2 na uwiano wa kawaida ni 3. --- Swali la 31: Kigezo cha 4 cha A.P. ni 13 na kigezo cha 7 ni 22. Tafuta jumla ya vigezo 10 vya kwanza. Fomula ya kigezo cha n cha A.P. ni a_n = a + (n-1)d. Fomula ya jumla ya vigezo n vya kwanza vya A.P. ni S_n = (n)/(2)[2a + (n-1)d]. Step 1: Andika milinganyo kutokana na taarifa uliyopewa. Kigezo cha 4: a_4 = a + (4-1)d = a + 3d = 13 (Mlinganyo 1) Kigezo cha 7: a_7 = a + (7-1)d = a + 6d = 22 (Mlinganyo 2) Step 2: Tatua kwa tofauti ya kawaida (d). Toa Mlinganyo 1 kutoka Mlinganyo 2: (a + 6d) - (a + 3d) = 22 - 13 3d = 9 d = (9)/(3) d = 3 Step 3: Tatua kwa kigezo cha kwanza (a). Badilisha d=3 katika Mlinganyo 1: a + 3(3) = 13 a + 9 = 13 a = 13 - 9 a = 4 Step 4: Tafuta jumla ya vigezo 10 vya kwanza (S_10). Tumia fomula S_n = (n)/(2)[2a + (n-1)d] na n=10, a=4, na d=3. S_10 = (10)/(2)[2(4) + (10-1)3] S_10 = 5[8 + (9)3] S_10 = 5[8 + 27] S_10 = 5[35] S_10 = 175 Jumla ya vigezo 10 vya kwanza ni 175. That's 2 down. 3 left today — send the next one.