7.20
[tex]a_n=a_1+(n-1)*r\\\\a) \\a_1 = -3\\r = 3\\a_6 = -3+(6-1)*3\\a_6=-3+15\\a_6=12\\\\a_{12}=-3+(12-1)*3\\a_{12}=-3+33\\a_{12}=30[/tex]
[tex]b)\\a_5=8\\r=-2\\a_5=a_1+4r\\8=a_1+4*(-2)\\8=a_1-8\\16=a_1\\\\a_{100}=16+99*(-2)\\a_{100}=16-198\\a_{100} = -182[/tex]
7.5
[tex]a) \\a_n=n^2-1\\a_1 = 1^2-1\\a_1=1-1\\a_1=0\\\\a_2=2^2-1\\a_2=4-1\\a_2=3\\\\a_3=3^2-1\\a_3=9-1\\a_3=8\\\\a_4=4^2-1\\a_4=16-1\\a-4=15\\\\a_5=5^2-1\\a_5=25-1\\a_5=24[/tex]
[tex]b) \\a_n=(-1)^n*2^{2n}\\\\a_1=(-1)^1*2^{2*1}\\a_1=-1*2^2\\a_1=-1*4\\a_1=-4\\\\a_2=(-1)^2*2^{2*2}\\a_2=1*2^4\\a_2=1*16\\a_2=16\\\\a_3=(-1)^3*2^{2*3}\\a_3=-1*2^6\\a_3=-1*64\\a_3=-64\\\\a_4 = (-1)^4*2^{2*4}\\a_4=1*2^8\\a_4=1*256\\a_4=256\\\\a_5=(-1)^5*2^{2*5}\\a_5=-1*2^{10}\\a_5=-1*1024\\a_5=-1024[/tex]
7.56
[tex]a_n=a_1*q^{n-1}[/tex]
a)
[tex]a_1=\sqrt2\\a_2=\sqrt6\\a_3=3\sqrt2\\a_2=a_1*q^{1}\\a_2=a_1*q\\\sqrt6=\sqrt2*q /:{\sqrt2}\\\frac{\sqrt6}{\sqrt2} = q\\\sqrt{\frac62}=q\\q=\sqrt3\\\\a_7=\sqrt2*(\sqrt3)^{6}\\a_7=\sqrt2*\sqrt{3^6}\\a_7=\sqrt2*\sqrt{729}\\a_7=\sqrt2*27\\a_7=27\sqrt2\\\\a_{14} = \sqrt2*(\sqrt3)^{13}\\a_{14}=\sqrt{2}*\sqrt{3^{13}}\\a_{14}=\sqrt2*\sqrt{3^6*3^6*3}\\a_{14}=\sqrt2*27\sqrt3\\a_{14}=27\sqrt6[/tex]
b)
[tex]a_1=\frac23\\a_2=1\\a_3=\frac32\\a_2=a_1*q\\1=\frac23*q /*\frac32\\\frac32=q\\\\a_5=\frac23*(\frac32)^4\\a_5=\frac23*\frac{81}{16}\\a_5=\frac{27}{8}\\a_5=3\frac38[/tex]
[tex]a_7 = \frac23*(\frac32)^6\\a_7=(\frac32)^{-1} * (\frac32)^6\\a_7=(\frac32)^5\\a_7=\frac{243}{32}\\a_7=7\frac{19}{32}[/tex]
c)
[tex]a_1 = \frac1{\sqrt2}\\a_2=1\\a_3=\sqrt2\\1=\frac1{\sqrt2}*q /*\sqrt2\\\sqrt2=q\\\\a_4 = \frac1{\sqrt2}*(\sqrt2)^3\\a_4=\frac1{\sqrt2}*\sqrt{2^3}\\a_4=\frac1{\sqrt2}*\sqrt8\\a_4=\frac1{\sqrt2}*2\sqrt2\\a_4=2\\\\a_8=\frac1{\sqrt2}*(\sqrt2)^7\\a_8=\frac1{\sqrt2}*\sqrt{2^7}\\a_8=\frac1{\sqrt2}*\sqrt{2^3*2^3*2}\\a_8=\frac1{\sqrt2}*8\sqrt2\\a_8=8[/tex]
d)
[tex]a_1=\frac1{64}\\a_2=\frac1{32}\\a_3=\frac1{16}\\\frac1{32}=\frac1{64}*q /*64\\\frac{64}{32}=q\\2=q\\\\a_5=\frac1{64}*2^4\\a_5=\frac1{64}*16\\a_5=\frac{16}{64}=\frac{2}{8}=\frac{1}{4}\\a_5=\frac14\\\\a_{10}=\frac1{64}*2^9\\a_{10} = \frac1{2^6}*2^9\\a_{10}=1*2^3\\a_{10}=8[/tex]
