The ratio of the sum of the first 21 terms to the sum of the first 14 terms = 63/60
Sum of the first n terms of a geometric series is:
[tex]S_n= \frac{a(r^n-1)}{r-1}[/tex]
Sum of the first seven terms of the geometric series
[tex]S_7= \frac{a(r^7-1)}{r-1}[/tex]
Sum of the first 14 terms of the geometric series
[tex]S_{14}= \frac{a(r^{14}-1)}{r-1}[/tex]
The sum of the first 7 terms is 4 times the sum of the following 7 terms.
[tex]S_7= 4(S_{14}-S_7)\\S_7= 4S_{14} - 4S_7\\5S_7 = 4S_{14}[/tex]
[tex]5 \frac{a(r^7-1)}{r-1} = 4 \frac{a(r^{14}-1)}{r-1}\\\\ \frac{5a(r^7-1)}{r-1} = \frac{4a(r^{14}-1)}{r-1}\\\\5a(r^7-1) = 4a(r^{14}-1)\\\\\frac{5}{4} = \frac{r^{14}-1}{r^7-1} \\\\\frac{5a}{4a} = \frac{r^{14}-1}{r^7-1}[/tex]
[tex]\frac{5}{4} = \frac{r^{14}-1}{r^7-1}\\\\5(r^7-1)=4(r^{14}-1)\\\\x = r^7\\5(x-1) = 4(x^2-1)\\\\4x^2-4=5x-5\\\\4x^2-5x-4+5=0\\\\4x^2-5x+1=0\\\\4x^2-x-4x+1=0\\\\x(4x-1)-1(4x-1)=0\\\\(x-1)(4x-1)=0\\\\x - 1 = 0\\\\x = 1\\\\4x-1=0\\\\4x = 1\\\\x = \frac{1}{4}[/tex]
[tex]\frac{1}{4} = r^7\\\\r = (\frac{1}{4})^{\frac{1}{7}[/tex]
Ratio of the sum of the first 21 terms to the sum of the first 14 terms
[tex]\frac{S_{21}}{S_{14}} = \frac{a(r^{21}-1)}{r-1} \div \frac{a(r^{14}-1)}{r-1}\\\\\frac{S_{21}}{S_{14}} = \frac{a(r^{21}-1)}{r-1} \times \frac{r-1}{a(r^{14}-1)}\\\\\frac{S_{21}}{S_{14}} =\frac{r^{21}-1}{r^{14}-1} \\\\\frac{S_{21}}{S_{14}} = \frac{(\frac{1}{4} )^{\frac{21}{7} }-1}{(\frac{1}{4} )^{\frac{14}{7} }-1}\\\\\frac{S_{21}}{S_{14}} =\frac{(\frac{1}{4} )^3-1}{(\frac{1}{4} )^2-1} \\\\\frac{S_{21}}{S_{14}} = \frac{(\frac{1}{64} )-1}{(\frac{1}{16} )-1} \\[/tex]
[tex]\frac{S_{21}}{S_{14}} =\frac{\frac{-63}{64} }{\frac{-15}{16} } \\\\\frac{S_{21}}{S_{14}} = \frac{-63}{64} \times \frac{16}{-15}\\\\\frac{S_{21}}{S_{14}} = \frac{63}{60}[/tex]
The ratio of the sum of the first 21 terms to the sum of the first 14 terms = 63/60
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