![\sqrt{x+7}+ \sqrt{x-2}=9](https://tex.z-dn.net/?f=+%5Csqrt%7Bx%2B7%7D%2B+%5Csqrt%7Bx-2%7D%3D9)
ОДЗ:
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{x+7≥0 {x≥-7 ---[-7]-------------------->
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{x-2≥0 {x≥2 -------------[2]----------->
x≥2
![(\sqrt{x+7}+ \sqrt{x-2})^2=9^2 \\ x+7+x-2+2 \sqrt{(x+7)(x-2)}=81 \\ 2 \sqrt{(x+7)(x-2)}=81-5-2x \\2 \sqrt{(x+7)(x-2)}=76-2x|:2 \\ \sqrt{(x+7)(x-2)}=38-x \\ \left \{ {{38-x \geq 0} \atop {(\sqrt{(x+7)(x-2)})^2=(38-x)^2}} \right. \\ \left \{ {{x \leq 38} \atop {x^2+7x-2x-14=1444+x^2-76x}} \right. \\ \left \{ {{x \leq 38} \atop {5x+76x=1444+14}} \right. \\ \left \{ {{x \leq 38} \atop {81x=1458}} \right. \\ \left \{ {{x \leq 38} \atop {x=18}} \right.](https://tex.z-dn.net/?f=%28%5Csqrt%7Bx%2B7%7D%2B+%5Csqrt%7Bx-2%7D%29%5E2%3D9%5E2+%5C%5C+x%2B7%2Bx-2%2B2+%5Csqrt%7B%28x%2B7%29%28x-2%29%7D%3D81+%5C%5C+2+%5Csqrt%7B%28x%2B7%29%28x-2%29%7D%3D81-5-2x+%5C%5C2+%5Csqrt%7B%28x%2B7%29%28x-2%29%7D%3D76-2x%7C%3A2+%5C%5C+%5Csqrt%7B%28x%2B7%29%28x-2%29%7D%3D38-x+%5C%5C+%5Cleft+%5C%7B+%7B%7B38-x+%5Cgeq+0%7D+%5Catop+%7B%28%5Csqrt%7B%28x%2B7%29%28x-2%29%7D%29%5E2%3D%2838-x%29%5E2%7D%7D+%5Cright.+%5C%5C+%5Cleft+%5C%7B+%7B%7Bx+%5Cleq+38%7D+%5Catop+%7Bx%5E2%2B7x-2x-14%3D1444%2Bx%5E2-76x%7D%7D+%5Cright.+%5C%5C+%5Cleft+%5C%7B+%7B%7Bx+%5Cleq+38%7D+%5Catop+%7B5x%2B76x%3D1444%2B14%7D%7D+%5Cright.+%5C%5C+%5Cleft+%5C%7B+%7B%7Bx+%5Cleq+38%7D+%5Catop+%7B81x%3D1458%7D%7D+%5Cright.+%5C%5C+%5Cleft+%5C%7B+%7B%7Bx+%5Cleq+38%7D+%5Catop+%7Bx%3D18%7D%7D+%5Cright.+)
Ответ: х=18
a) Используя n-ый член геометрической прогрессии
, найдем первый член этой прогрессии
![\rm b_1=\dfrac{b_n}{q^{n-1}}=\dfrac{b_4}{q^3}=\dfrac{3/64}{(1/2)^3}=\dfrac{3}{8}](https://tex.z-dn.net/?f=%5Crm%20b_1%3D%5Cdfrac%7Bb_n%7D%7Bq%5E%7Bn-1%7D%7D%3D%5Cdfrac%7Bb_4%7D%7Bq%5E3%7D%3D%5Cdfrac%7B3%2F64%7D%7B%281%2F2%29%5E3%7D%3D%5Cdfrac%7B3%7D%7B8%7D)
Сумма n-первых членов геометрической прогрессии вычисляется по следующей формуле:
, тогда сумма первых восьми членов этой прогрессии:
![\rm S_8=\dfrac{\dfrac{3}{8}\cdot \left[1-\bigg(\dfrac{1}{2}\bigg)^8\right]}{1-\dfrac{1}{2}}=\dfrac{3}{4}](https://tex.z-dn.net/?f=%5Crm%20S_8%3D%5Cdfrac%7B%5Cdfrac%7B3%7D%7B8%7D%5Ccdot%20%5Cleft%5B1-%5Cbigg%28%5Cdfrac%7B1%7D%7B2%7D%5Cbigg%29%5E8%5Cright%5D%7D%7B1-%5Cdfrac%7B1%7D%7B2%7D%7D%3D%5Cdfrac%7B3%7D%7B4%7D)
Ответ: 3/4.
б) Аналогично, найдем первый член геометрической прогрессии, используя n-ый член этой прогрессии:
![\rm b_1=\dfrac{b_n}{q^{n-1}}=\dfrac{b_5}{q^4}=\dfrac{-9}{(-3)^4}=-\dfrac{1}{9}](https://tex.z-dn.net/?f=%5Crm%20b_1%3D%5Cdfrac%7Bb_n%7D%7Bq%5E%7Bn-1%7D%7D%3D%5Cdfrac%7Bb_5%7D%7Bq%5E4%7D%3D%5Cdfrac%7B-9%7D%7B%28-3%29%5E4%7D%3D-%5Cdfrac%7B1%7D%7B9%7D)
Тогда сумма первых десяти членов геометрической прогрессии:
![\rm S_{10}=\dfrac{b_1\left(1-q^{10}\right)}{1-q}=-\dfrac{\dfrac{1}{9}\cdot \left[1-\big(-3\big)^{10}\right]}{1+3}=\dfrac{14762}{9}](https://tex.z-dn.net/?f=%5Crm%20S_%7B10%7D%3D%5Cdfrac%7Bb_1%5Cleft%281-q%5E%7B10%7D%5Cright%29%7D%7B1-q%7D%3D-%5Cdfrac%7B%5Cdfrac%7B1%7D%7B9%7D%5Ccdot%20%5Cleft%5B1-%5Cbig%28-3%5Cbig%29%5E%7B10%7D%5Cright%5D%7D%7B1%2B3%7D%3D%5Cdfrac%7B14762%7D%7B9%7D)
Ответ: 14762/9
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