Answer:
![\large\boxed{10.\ \sum\limits_{n=1}^{9}5(2)^{n-1}=2,555}](https://tex.z-dn.net/?f=%5Clarge%5Cboxed%7B10.%5C%20%5Csum%5Climits_%7Bn%3D1%7D%5E%7B9%7D5%282%29%5E%7Bn-1%7D%3D2%2C555%7D)
![\large\boxed{11.\ S_7=6,564}](https://tex.z-dn.net/?f=%5Clarge%5Cboxed%7B11.%5C%20S_7%3D6%2C564%7D)
![\large\boxed{12. a_1=160}](https://tex.z-dn.net/?f=%5Clarge%5Cboxed%7B12.%20a_1%3D160%7D)
Step-by-step explanation:
10.
The formula of a sum of terms of a geometric sequence:
![S_n=\dfrac{a_1(1-r^n)}{1-r}](https://tex.z-dn.net/?f=S_n%3D%5Cdfrac%7Ba_1%281-r%5En%29%7D%7B1-r%7D)
We have:
![\sum\limits_{n=1}^{9}5(2)^{n-1}](https://tex.z-dn.net/?f=%5Csum%5Climits_%7Bn%3D1%7D%5E%7B9%7D5%282%29%5E%7Bn-1%7D)
Therefore:
![a_n=5(2)^{n-1}\to a_1=5(2)^{1-1}=5(2)^0=5(1)=5](https://tex.z-dn.net/?f=a_n%3D5%282%29%5E%7Bn-1%7D%5Cto%20a_1%3D5%282%29%5E%7B1-1%7D%3D5%282%29%5E0%3D5%281%29%3D5)
![r=\dfrac{a_{n+1}}{a_n}\\\\a_{n+1}=5(2)^{n+1-1}=5(2)^n\\\\r=\dfrac{5(2)^n}{5(2)^{n-1}}=\dfrac{2^n}{2^{n-1}}\qquad\text{use}\ \dfrac{a^n}{a^m}=a^{n-m}\\\\r=2^{n-(n-1)}=2^{n-n+1}=2^1=2](https://tex.z-dn.net/?f=r%3D%5Cdfrac%7Ba_%7Bn%2B1%7D%7D%7Ba_n%7D%5C%5C%5C%5Ca_%7Bn%2B1%7D%3D5%282%29%5E%7Bn%2B1-1%7D%3D5%282%29%5En%5C%5C%5C%5Cr%3D%5Cdfrac%7B5%282%29%5En%7D%7B5%282%29%5E%7Bn-1%7D%7D%3D%5Cdfrac%7B2%5En%7D%7B2%5E%7Bn-1%7D%7D%5Cqquad%5Ctext%7Buse%7D%5C%20%5Cdfrac%7Ba%5En%7D%7Ba%5Em%7D%3Da%5E%7Bn-m%7D%5C%5C%5C%5Cr%3D2%5E%7Bn-%28n-1%29%7D%3D2%5E%7Bn-n%2B1%7D%3D2%5E1%3D2)
Substitute a₁ = 5, r = 2 and n = 9:
![S_9=\dfrac{5(1-2^9)}{1-2}=\dfrac{5(1-512)}{-1}=-5(-511)=2,555](https://tex.z-dn.net/?f=S_9%3D%5Cdfrac%7B5%281-2%5E9%29%7D%7B1-2%7D%3D%5Cdfrac%7B5%281-512%29%7D%7B-1%7D%3D-5%28-511%29%3D2%2C555)
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11.
We have
![a_2=-36,\ a_5=972,\ n=7](https://tex.z-dn.net/?f=a_2%3D-36%2C%5C%20a_5%3D972%2C%5C%20n%3D7)
We know:
![a_n=a_1r^{n-1}](https://tex.z-dn.net/?f=a_n%3Da_1r%5E%7Bn-1%7D)
Therefore
![a_2=a_1r^{2-1}=a_1r^1=a_1r\\\\a_5=a_1r^{5-1}=a_1r^4\\\\\dfrac{a_5}{a_2}=\dfrac{a_1r^4}{a_1r}=r^{4-1}=r^3](https://tex.z-dn.net/?f=a_2%3Da_1r%5E%7B2-1%7D%3Da_1r%5E1%3Da_1r%5C%5C%5C%5Ca_5%3Da_1r%5E%7B5-1%7D%3Da_1r%5E4%5C%5C%5C%5C%5Cdfrac%7Ba_5%7D%7Ba_2%7D%3D%5Cdfrac%7Ba_1r%5E4%7D%7Ba_1r%7D%3Dr%5E%7B4-1%7D%3Dr%5E3)
Substitute:
![r^3=\dfrac{972}{-36}\\\\r^3=-27\to r=\sqrt[3]{-27}\\\\r=-3](https://tex.z-dn.net/?f=r%5E3%3D%5Cdfrac%7B972%7D%7B-36%7D%5C%5C%5C%5Cr%5E3%3D-27%5Cto%20r%3D%5Csqrt%5B3%5D%7B-27%7D%5C%5C%5C%5Cr%3D-3)
Calculate the first term:
![a_2=a_1r\to a_1=\dfrac{a_2}{r}\\\\a_1=\dfrac{-36}{-3}=12](https://tex.z-dn.net/?f=a_2%3Da_1r%5Cto%20a_1%3D%5Cdfrac%7Ba_2%7D%7Br%7D%5C%5C%5C%5Ca_1%3D%5Cdfrac%7B-36%7D%7B-3%7D%3D12)
Put a₁ = 12, r = -3 and n = 7 to the formula of a sum:
![S_7=\dfrac{12(1-(-3)^7)}{1-(-3)}=\dfrac{12(1-(-2187))}{1+3}=\dfrac{12(1+2187)}{4}=3(2188)\\\\S_7=6564](https://tex.z-dn.net/?f=S_7%3D%5Cdfrac%7B12%281-%28-3%29%5E7%29%7D%7B1-%28-3%29%7D%3D%5Cdfrac%7B12%281-%28-2187%29%29%7D%7B1%2B3%7D%3D%5Cdfrac%7B12%281%2B2187%29%7D%7B4%7D%3D3%282188%29%5C%5C%5C%5CS_7%3D6564)
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12.
We have
![n=6,\ S_n=315\to S_6=315,\ a_n=5\to a_6=5,\ r=\dfrac{1}{2}](https://tex.z-dn.net/?f=n%3D6%2C%5C%20S_n%3D315%5Cto%20S_6%3D315%2C%5C%20a_n%3D5%5Cto%20a_6%3D5%2C%5C%20r%3D%5Cdfrac%7B1%7D%7B2%7D)
We know:
![a_n=a_1r^{n-1}\to a_6=a_1r^{6-1}=a_1r^5](https://tex.z-dn.net/?f=a_n%3Da_1r%5E%7Bn-1%7D%5Cto%20a_6%3Da_1r%5E%7B6-1%7D%3Da_1r%5E5)
Substitute:
![5=a_1\left(\dfrac{1}{2}\right)^5\\\\5=\dfrac{1}{32}a_1\qquad\text{multiply both sides by 32}\\\\160=a_1\to a_1=160](https://tex.z-dn.net/?f=5%3Da_1%5Cleft%28%5Cdfrac%7B1%7D%7B2%7D%5Cright%29%5E5%5C%5C%5C%5C5%3D%5Cdfrac%7B1%7D%7B32%7Da_1%5Cqquad%5Ctext%7Bmultiply%20both%20sides%20by%2032%7D%5C%5C%5C%5C160%3Da_1%5Cto%20a_1%3D160)
Check for the given sum:
Substitute a₁ = 160, r = 1/2 and n = 6:
![S_6=\dfrac{160\left(1-\left(\frac{1}{2}\right)^6\right)}{1-\frac{1}{2}}=\dfrac{160\left(1-\frac{1}{64}\right)}{\frac{1}{2}}=160\left(\dfrac{64}{64}-\dfrac{1}{64}\right)\left(\dfrac{2}{1}\right)\\\\=160\left(\dfrac{63}{64}\right)(2)=(2.5)(63)(2)=315](https://tex.z-dn.net/?f=S_6%3D%5Cdfrac%7B160%5Cleft%281-%5Cleft%28%5Cfrac%7B1%7D%7B2%7D%5Cright%29%5E6%5Cright%29%7D%7B1-%5Cfrac%7B1%7D%7B2%7D%7D%3D%5Cdfrac%7B160%5Cleft%281-%5Cfrac%7B1%7D%7B64%7D%5Cright%29%7D%7B%5Cfrac%7B1%7D%7B2%7D%7D%3D160%5Cleft%28%5Cdfrac%7B64%7D%7B64%7D-%5Cdfrac%7B1%7D%7B64%7D%5Cright%29%5Cleft%28%5Cdfrac%7B2%7D%7B1%7D%5Cright%29%5C%5C%5C%5C%3D160%5Cleft%28%5Cdfrac%7B63%7D%7B64%7D%5Cright%29%282%29%3D%282.5%29%2863%29%282%29%3D315)
CORRECT :)