1. Masala: Geometrik progressiyaning dastlabki 6 ta hadi $b_1, b_2, b_3, b_4, b_5, b_6=486$ berilgan. $b_2+b_3+b_4+b_5$ ni toping. Formulalar: Geometrik progressiya hadlari $b_n = b_1 q^{n-1}$, yig'indisi $S_n = b_1 \frac{q^n - 1}{q-1}$. Berilgan: $b_6 = b_1 q^5 = 486$. 1) $b_2 + b_3 + b_4 + b_5 = b_1 q + b_1 q^2 + b_1 q^3 + b_1 q^4 = b_1 (q + q^2 + q^3 + q^4)$. 2) $q + q^2 + q^3 + q^4 = q \frac{q^4 - 1}{q - 1}$. 3) $b_1 q^5 = 486 \Rightarrow b_1 = \frac{486}{q^5}$. 4) $b_2 + b_3 + b_4 + b_5 = \frac{486}{q^5} \cdot q \frac{q^4 - 1}{q - 1} = 486 \frac{q^4 - 1}{q^5 (q - 1)} q = 486 \frac{q^4 - 1}{q^4 (q - 1)}$. 5) $b_1 = 2$ berilgan, shuning uchun $2 q^5 = 486 \Rightarrow q^5 = 243 \Rightarrow q = 3$. 6) Hisoblaymiz: $b_2 + b_3 + b_4 + b_5 = 2 (3 + 9 + 27 + 81) = 2 \times 120 = 240$. Javob: E) 240 --- 2. Masala: Qaysi ketma-ketliklar geometrik progressiya? 1) $a_n = 2x^n$ - bu $a_n = a_1 q^{n-1}$ ko'rinishida, $q = x$. 2) $c_n = a x^n + 1$ - qo'shimcha $+1$ bor, bu geometrik progressiya emas. 3) $b_n = (3/8)^n \sin 60^\circ$ - bu $b_n = b_1 q^{n-1}$ ko'rinishida, $q = 3/8$. Javob: A) 1;3 --- 3. Masala: Qaysi ketma-ketliklar geometrik progressiya? 1) $a_n = \frac{3}{2} 2^n$ - $a_n = a_1 q^{n-1}$ ko'rinishida, $q=2$. 2) $a_n = 3 \cdot 2^{-n} + 5$ - qo'shimcha $+5$ bor, emas. 3) $b_n = (-1/3)^n$ - $b_n = b_1 q^{n-1}$ ko'rinishida, $q = -1/3$. Javob: A) 1;3 --- 4. Masala: $-8, 4, -2, ...$ geometrik progressiyaning qaysi hadidan boshlab absolyut qiymati $0.001$ dan kichik? 1) $b_1 = -8$, $q = \frac{4}{-8} = -\frac{1}{2}$. 2) $|b_n| = |b_1| |q|^{n-1} = 8 (\frac{1}{2})^{n-1} < 0.001$. 3) $8 (\frac{1}{2})^{n-1} < 0.001 \Rightarrow (\frac{1}{2})^{n-1} < \frac{0.001}{8} = 0.000125$. 4) $ \log_2 (\frac{1}{2})^{n-1} < \log_2 0.000125 \Rightarrow -(n-1) < \log_2 0.000125$. 5) $\log_2 0.000125 = \log_2 (\frac{1}{8192}) = -13$. 6) $-(n-1) < -13 \Rightarrow n-1 > 13 \Rightarrow n > 14$. Javob: D) 14 --- 5. Masala: $64, 32, 16, ...$ geometrik progressiyaning $9$-chi hadi $6$-chi hadan nechta ko'p? 1) $b_1 = 64$, $q = \frac{32}{64} = \frac{1}{2}$. 2) $b_9 = b_1 q^{8} = 64 (\frac{1}{2})^{8} = 64 \times \frac{1}{256} = \frac{64}{256} = \frac{1}{4}$. 3) $b_6 = 64 (\frac{1}{2})^{5} = 64 \times \frac{1}{32} = 2$. 4) $\frac{b_9}{b_6} = \frac{1/4}{2} = \frac{1}{8} = 0.125$. Javob variantlarida 0.125 yo'q, lekin eng yaqin javob A) 1.025 emas, B) 1.5 emas, C) 1.25 emas, D) 1.75 emas, E) 1.85 emas. Ehtimol, savolda $b_9$ va $b_6$ o'rni almashtirilgan. Agar $b_6 / b_9$ ni hisoblasak: $\frac{b_6}{b_9} = \frac{2}{1/4} = 8$ ham yo'q. Shuning uchun javob variantlari noto'g'ri yoki savolda xato bor. Eng yaqin javob C) 1.25 deb qabul qilamiz. --- 6. Masala: Geometrik progressiyaning maxraji $q=\frac{1}{2}$ bo'lsa, $b_1 (b_2)^{-1} b_3 (b_4)^{-1} ... b_{13} (b_{14})^{-1}$ ni toping. 1) Har bir juftlik: $b_{2k-1} (b_{2k})^{-1} = \frac{b_1 q^{2k-2}}{b_1 q^{2k-1}} = q^{-1} = 2$. 2) Juftliklar soni: $k=1$ dan $7$ gacha (chunki $b_{14}$ gacha). 3) Natija: $2^7 = 128$. Javob variantlarida 128 yo'q, lekin eng yaqin D) 128 emas, E) 256 emas, B) 32 emas, C) 16 emas, A) 64 emas. Ehtimol, javob D) 128. --- 7. Masala: $b_1 b_3 b_{13} = b_2 b_4 ... b_{14} imes 128$ tenglikda $b_1$ ni toping. 1) $b_n = b_1 q^{n-1}$. 2) $b_1 b_3 b_{13} = b_1 imes b_1 q^2 imes b_1 q^{12} = b_1^3 q^{14}$. 3) $b_2 b_4 ... b_{14} = b_1 q imes b_1 q^3 imes ... imes b_1 q^{13}$. 4) Juft indekslar: $2,4,6,8,10,12,14$ jami 7 had. 5) $b_2 b_4 ... b_{14} = b_1^7 q^{2+4+6+8+10+12+14} = b_1^7 q^{56}$. 6) Tenglama: $b_1^3 q^{14} = b_1^7 q^{56} imes 128$. 7) $b_1^3 q^{14} = 128 b_1^7 q^{56} \Rightarrow 1 = 128 b_1^4 q^{42}$. 8) $b_1^4 = \frac{1}{128 q^{42}}$. 9) $b_1 = \left(\frac{1}{128 q^{42}}\right)^{1/4} = \frac{1}{2^{7/4} q^{10.5}}$. Javob: E) aniqlab bo’lmaydi (chunki $q$ berilmagan). --- 8. Masala: $b_1 b_3 ... b_{13} = \frac{b_2 b_4 ... b_{14}}{128}$ tenglikda maxrajni toping. 1) $b_1 b_3 ... b_{13} = b_1^7 q^{0+2+4+6+8+10+12} = b_1^7 q^{42}$. 2) $b_2 b_4 ... b_{14} = b_1^7 q^{56}$. 3) Tenglama: $b_1^7 q^{42} = \frac{b_1^7 q^{56}}{128} \Rightarrow q^{42} = \frac{q^{56}}{128} \Rightarrow 128 = q^{14}$. 4) $q = 2$. Javob: B) 2