document.write( "Question 172798: For a geometric sequence with $\"r=1.5\"$ and $\"S%5B5%5D=92.3125\"$ . Find $\"S%5B10%5D\"$ \n" ); document.write( "

Algebra.Com's Answer #127640 by jim_thompson5910(35256) $\"\"$ $\"About$

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Note: $\"S%5B5%5D\"$ is the 5th partial sum of the geometric sequence. To find the 5th partial sum, we use the value $\"n=4\"$ (since we start at $\"n=0\"$ ) and use the formula\r
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\n" ); document.write( "\n" ); document.write( " $\"sum%28a%2Ar%5Ek%2Ck=1%2Cn%29=%28a%281-r%5E%28n%2B1%29%29%29%2F%281-r%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Since we know that $\"S%5B5%5D=92.3125\"$ , this means that $\"sum%28a%2A%281.5%29%5En%2Cn=1%2C5%29=92.3125\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"92.3125=%28a%281-r%5E%28n%2B1%29%29%29%2F%281-r%29\"$ Replace the left side with 92.3125 \r
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\n" ); document.write( "\n" ); document.write( " $\"92.3125=%28a%281-%281.5%29%5E%284%2B1%29%29%29%2F%281-1.5%29\"$ Plug in $\"r=1.5\"$ and $\"n=4\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"92.3125=%28a%281-%281.5%29%5E%285%29%29%29%2F%28-0.5%29\"$ Combine like terms.\r
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\n" ); document.write( "\n" ); document.write( " $\"92.3125=%28a%281-7.59375%29%29%2F%28-0.5%29\"$ Raise 1.5 to the 5th power to get 7.59375\r
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\n" ); document.write( "\n" ); document.write( " $\"92.3125=%28a%28-6.59375%29%29%2F%28-0.5%29\"$ Combine like terms.\r
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\n" ); document.write( "\n" ); document.write( " $\"92.3125=13.1875a\"$ Divide\r
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\n" ); document.write( "\n" ); document.write( " $\"92.3125%2F13.1875=a\"$ Divide both sides by 13.1875 to isolate \"a\"\r
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\n" ); document.write( "\n" ); document.write( " $\"7=a\"$ Divide\r
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\n" ); document.write( "\n" ); document.write( "So we find that $\"a=7\"$ (which is the first term)\r
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\n" ); document.write( "\n" ); document.write( "Now to find $\"S%5B10%5D\"$ , we will use $\"n=9\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"%28a%281-r%5E%28n%2B1%29%29%29%2F%281-r%29\"$ Start with the right side of the formula\r
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\n" ); document.write( "\n" ); document.write( " $\"%287%281-%281.5%29%5E%289%2B1%29%29%29%2F%281-1.5%29\"$ Plug in $\"a=7\"$ , $\"r=1.5\"$ , and $\"n=9\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"%287%281-%281.5%29%5E%2810%29%29%29%2F%28-0.5%29\"$ Combine like terms.\r
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\n" ); document.write( "\n" ); document.write( " $\"%287%281-57.665%29%29%2F%28-0.5%29\"$ Raise 1.5 to the 10th power to get 57.665 (this is approximate)\r
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\n" ); document.write( "\n" ); document.write( " $\"%287%28-56.665%29%29%2F%28-0.5%29\"$ Combine like terms.\r
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\n" ); document.write( "\n" ); document.write( " $\"%28-396.655%29%2F%28-0.5%29\"$ Multiply\r
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\n" ); document.write( "\n" ); document.write( " $\"793.31\"$ Divide\r
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\n" ); document.write( "\n" ); document.write( "So the tenth partial sum is approximately $\"793.31\"$ . This means that $\"S%5B10%5D=793.31\"$
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