document.write( "Question 64422: Use the geometric sequence of numbers 1, 1/2, 1/4, 1/8,...to find the following:\r
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\n" ); document.write( "\n" ); document.write( "b) Using the formula for the sum of the first n terms of a geometric series, what is the sum of the first 10 terms? Please round your answer to 4 decimals.\r
\n" ); document.write( "\n" ); document.write( "c) Using the formula for the sum of the first n terms of a geometric series, what is the sum of the first 12 terms? Please round your answer to 4 decimals.\r
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Algebra.Com's Answer #45042 by Edwin McCravy(20081)\"\" \"About 
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document.write( "Use the geometric sequence of numbers \r\n" );
document.write( "1, 1/2, 1/4, 1/8,...to \r\n" );
document.write( "find the following:\r\n" );
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document.write( "a) What is r, the ratio between 2 consecutive terms? \r\n" );
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document.write( "Just divide each given term  after the first by the \r\n" );
document.write( "preceding one and see if you get the same number.  If \r\n" );
document.write( "you do, then you call that number \"the common ratio, r\".\r\n" );
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document.write( "For 1, 1/2, 1/4, 1/8,... we divide the second term, 1/2,\r\n" );
document.write( "by the first term 1, like this: 1/2÷1 = 1/2.  Then we \r\n" );
document.write( "divide the third term 1/4, by the second term 1/2, like \r\n" );
document.write( "this: (1/4)÷(1/2) = (1/4)×(2/1) = 2/4 = 1/2. \r\n" );
document.write( "Then we divide the fourth term, 1/8, by the third term, \r\n" );
document.write( "1/4, like this\" (1/8)÷(1/4) = (1/8)×(4/1) = 4/8 = 1/2.  \r\n" );
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document.write( "Every time we got 1/2. So that means this is a geometric \r\n" );
document.write( "sequence and the common ratio, r, is 1/2.  So r = 1/2. \r\n" );
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document.write( "b) Using the formula for the sum of the first n \r\n" );
document.write( "terms of a geometric series, what is the sum of \r\n" );
document.write( "the first 10 terms? Please round your answer to \r\n" );
document.write( "4 decimals.\r\n" );
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document.write( "The formula for the sum, called Sn, of the \r\n" );
document.write( "first n terms of a geometric sequence is either \r\n" );
document.write( "of these two equivalent formulas:\r\n" );
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document.write( "Sn = a1(rn - 1)/(r - 1)\r\n" );
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document.write( "or\r\n" );
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document.write( "Sn = a1(1 - rn)/(1 - r)\r\n" );
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document.write( "where a1 stands for the first term, r stands \r\n" );
document.write( "for the common ratio, and n stands for the number of \r\n" );
document.write( "term that you want to find.  It doesn't matter which of \r\n" );
document.write( "those formulas you use, because you'll get the same \r\n" );
document.write( "answer using either one. Normally we use the first one \r\n" );
document.write( "if |r| > 1 and the second one if |r| < 1, but there is \r\n" );
document.write( "no rule. I'll use the second one. \r\n" );
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document.write( "Here a1 = 1, r = 1/2, and n = 10 so we plug those in:\r\n" );
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document.write( "Sn = a1(1 - rn)/(1 - r)\r\n" );
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document.write( "S10 = (1)[1 - (1/2)10]/(1 - 1/2)\r\n" );
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document.write( "S10 = (1)(1 - 1/210)/(1/2)\r\n" );
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document.write( "S10 = (1 - 1/210)/(1/2)\r\n" );
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document.write( "S10 = (1 - 1/210)×(2/1)\r\n" );
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document.write( "S10 = 2(1 - 1/210)\r\n" );
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document.write( "S10 = 2 - 2/210\r\n" );
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document.write( "S10 = 2 - 1/29 \r\n" );
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document.write( "S10 = 2 - 1/512 = 1.9980 \r\n" );
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document.write( "c) Using the formula for the sum of the first n \r\n" );
document.write( "terms of a geometric series, what is the sum of \r\n" );
document.write( "the first 12 terms? Please round your answer to \r\n" );
document.write( "4 decimals.\r\n" );
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document.write( "Same as above using 12 for n instead of 10\r\n" );
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document.write( "a1 = 1, r = 1/2, and n = 12 so we plug those in:\r\n" );
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document.write( "Sn = a1(1 - rn)/(1 - r)\r\n" );
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document.write( "S12 = (1)[1 - (1/2)12]/(1 - 1/2)\r\n" );
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document.write( "S12 = (1)(1 - 1/212)/(1/2)\r\n" );
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document.write( "S12 = (1 - 1/212)/(1/2)\r\n" );
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document.write( "S12 = (1 - 1/212)×(2/1)\r\n" );
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document.write( "S12 = 2(1 - 1/212)\r\n" );
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document.write( "S12 = 2 - 2/212\r\n" );
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document.write( "S12 = 2 - 1/211 \r\n" );
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document.write( "S12 = 2 - 1/2048 = 1.9995 \r\n" );
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document.write( "d) What observation can you make about these \r\n" );
document.write( "sums? In particular, what whole number does \r\n" );
document.write( "it appear that the sum will always be smaller \r\n" );
document.write( "than?\r\n" );
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document.write( "Well, we found:\r\n" );
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document.write( "S10 = 2 - 1/512 = 1.9980 \r\n" );
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document.write( "S12 = 2 - 1/2048 = 1.9995 \r\n" );
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document.write( "It appears that each time we will be\r\n" );
document.write( "subtracting a smaller fraction away\r\n" );
document.write( "from 2, so the whole number that the\r\n" );
document.write( "it appears the sum will always be a\r\n" );
document.write( "little smaller than -- is 2.\r\n" );
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document.write( "Edwin

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