[latex]a)\\1+2+4+8+16+32+64 = 127[/latex] [latex]S_{n} = a_1*frac{1-q^{n}}{1-q}\\a_1 = 1\q = frac{2}{1} = 2\n = 7\\S_{7} = 1*frac{1-2^{7}}{1-2} = frac{1-128}{-1} = 127[/latex] [latex]b)\\1-3+9-27+81-243+729 = 547[/latex] [latex]a_1 = 1\q = frac{-3}{1} = -3\n = 7\\S_{7} = 1*frac{1-(-3)^{7}}{1-(-3)} = frac{1+2187}{4} = frac{2188}{4} = 547[/latex] [latex]c)\\32-16+8-4+2-1+0,5-0,25 = 21,25[/latex] [latex]a_1 = 32\q = frac{-16}{32} = -frac{1}{2}\n = 8\\S_{8} = 32*frac{1-(-frac{1}{2})^{8}}{1-(-frac{1}{2})}= 32*frac{1-frac{1}{256}}{1frac{1}{2}}=32*frac{255}{256}:frac{3}{2} = 21,25[/latex]
A)1+2+4+...+64 1+2+4+8+16+32+64 = 31+96=127 [latex]a_1=1 \ q=a_3:a_2=4:2=2 \ a_n = a_1*q^{n-1} \ \ 64=1*2^{n-1} \ 2^6=2^{n-1} \ n-1=6 \ n=7 \ \ S_n=a_1* frac{1-q^n}{1-q} \ \ S_7=1* frac{1-2^7}{1-2} = frac{-127}{-1}=127 [/latex] B)1-3+9-27+81-243+729 1-3+9-27+81-243+729=10-30+810-243=820-273=547 [latex]a_1=1 \ q=a_3:a_2=9:(-3)=-3 \ a_n= a_1*q^{n-1} \ \ 729=1*(-3)^{n-1} \ (-3)^6=(-3)^{n-1} \ n-1=6 \ n=7 \ \ S_7= 1*frac{1-(-3)^7}{1+3}= frac{1+2187}{4}=547 [/latex] C). 32-16+8-4+2-1+0,5-0,25 32-16+8-4+2-1+0,5-0,25=16+4+1+0,25=21,25 [latex]a_1=32 \ q= frac{a_3}{a_1} = frac{8}{-16}=- frac{1}{2} \ a_n=a_1*q^{n-1} \ \ -0,25=32*(- frac{1}{2})^{n-1} \ \ - frac{1}{4} =32*(-2)^{n-1} /:32\ \ -frac{1}{128} =(-2)^{n-1} \ \ -2^{7} =(-2)^{n-1} \ \ (-2)^{7} =(-2)^{n-1}[/latex] [latex]n-1=7 \ \ n=8 \ \ S_8 = 32* frac{ig {1-( -frac{1}{2})^8} }{ig {1-( -frac{1}{2})}} =32*frac{ig {1-frac{1}{256}} }{ig {1+frac{1}{2}}} =32*frac{ig {frac{255}{256}} }{ig {frac{3}{2}}} =32*frac{255}{256}* frac{2}{3} =frac{85}{4}= \ \ =21frac{1}{4}=21,25[/latex]
