document.write( "Question 1201412: In a geometric series the seventh term is 1458, the ninth terms 13122. Find the second term algebraically \n" ); document.write( "
Algebra.Com's Answer #835765 by Theo(13342)\"\" \"About 
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the formula for the geometric series is An = A1 * r ^ (n-1)
\n" ); document.write( "An is the nth term.
\n" ); document.write( "A1 is the first term
\n" ); document.write( "r ^ (n-1) is the common multiplier raised to the (n-1)th term.
\n" ); document.write( "in your problem, the seventh term would be A7 and the ninth term would be 9
\n" ); document.write( "you will have:
\n" ); document.write( "A7 = 1458
\n" ); document.write( "A9 = 13122
\n" ); document.write( "the formula for the 7th term is A7 = A1 * r ^ 6
\n" ); document.write( "the formula for the 9th term is A9 = A1 * r ^ 8
\n" ); document.write( "replace A7 and A9 with their respective values to get:
\n" ); document.write( "1458 = A1 * r ^ 6
\n" ); document.write( "13122 = A1 * r ^ 8
\n" ); document.write( "solve for A1 in both equations to get:
\n" ); document.write( "A1 = 1458 / r^6
\n" ); document.write( "A1 = 13122 / r^8
\n" ); document.write( "since they are both equal to A1, you get:
\n" ); document.write( "1458 / r^6 = 13122 / r^8
\n" ); document.write( "multiply both sides of the equation by r^8 and divide both sides of the eqution by 1458 to get:
\n" ); document.write( "r^8 / r^6 = 13122 / 1458
\n" ); document.write( "simplify to get:
\n" ); document.write( "r^2 = 9
\n" ); document.write( "solve for r to get:
\n" ); document.write( "r = 3
\n" ); document.write( "replace r in the first equation to get:
\n" ); document.write( "1458 = A1 * 3 ^ 6
\n" ); document.write( "solve for A1 to get:
\n" ); document.write( "A1 = 1458 / 3^6 = 2
\n" ); document.write( "replace A1 with 2 in the second equation to get:
\n" ); document.write( "confirm by replacing A1 in the first eqution to get:
\n" ); document.write( "1458 = 2 * 3 ^ 6 = 1458.
\n" ); document.write( "do the same in the secone equation to get:
\n" ); document.write( "13122 = 2 * 3 ^ 8 = 13122.
\n" ); document.write( "you have solved for A1 and r.
\n" ); document.write( "A1 is 2 and r is 3.
\n" ); document.write( "if you want to find the second term algebraically, you would get:
\n" ); document.write( "A2 = 2 * 3^1 = 6
\n" ); document.write( "that's your solution.\r
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\n" ); document.write( "\n" ); document.write( "you could also have solved as follows:
\n" ); document.write( "A9 / A7 = (A1 * r^8)/A1 * r^6)
\n" ); document.write( "the A1 in the numerator and denominator cancel out and you are left with:
\n" ); document.write( "A9 / A7 = r^8 / r^6
\n" ); document.write( "simplify to get A9 / A7 = r^2
\n" ); document.write( "this becomes 13122 / 1458 = r^2
\n" ); document.write( "solve for r to get:
\n" ); document.write( "r = sqrt(13122 / 1458) = 3.
\n" ); document.write( "once you found r, then you can find A1 by just replacing r in the equation and solving for A1.
\n" ); document.write( "for example:
\n" ); document.write( "1458 = A1 * 3^6
\n" ); document.write( "this becomes 1458 = A1 * 749
\n" ); document.write( "solve for A1 to get:
\n" ); document.write( "A1 = 1458 / 729 = 2
\n" ); document.write( "you now have:
\n" ); document.write( "r = 3
\n" ); document.write( "A1 = 2
\n" ); document.write( "from there, you can easily find A2 because A2 = A1 * r ^ (2-1) which becomes A2 = 2 * 3 ^ 1 which becomes A2 = 2 * 3 which is equal to 6.
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