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 #835768 by math_tutor2020(3817)\"\" \"About 
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\n" ); document.write( "r = common ratio, some nonzero value
\n" ); document.write( "a7 = 7th term = 1458
\n" ); document.write( "a8 = 8th term = a7*r = 1458r
\n" ); document.write( "a9 = 9th term = a8*r = (1458r)*r = 1458r^2
\n" ); document.write( "Each time we need a new term, we multiply the previous term by r.\r
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\n" ); document.write( "\n" ); document.write( "Set the 9th term equal to the stated value 13122 and solve for r.
\n" ); document.write( "1458r^2 = 13122
\n" ); document.write( "r^2 = 13122/1458
\n" ); document.write( "r^2 = 9
\n" ); document.write( "r = sqrt(9) or r = -sqrt(9)
\n" ); document.write( "r = 3 or r = -3
\n" ); document.write( "We have two possibilities for r.
\n" ); document.write( "The negative r value will make the geometric sequence terms bounce around from positive to negative, or vice versa.\r
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\n" ); document.write( "\n" ); document.write( "Consider the case r = 3.
\n" ); document.write( "an = a1*(r)^(n-1)
\n" ); document.write( "a7 = a1*(3)^(7-1)
\n" ); document.write( "1458 = a1*(3)^6
\n" ); document.write( "a1 = 1458/(3^6)
\n" ); document.write( "a1 = 2\r
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\n" ); document.write( "\n" ); document.write( "One possible geometric sequence nth term formula is
\n" ); document.write( "an = 2*(3)^(n-1)\r
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\n" ); document.write( "\n" ); document.write( "Check:
\n" ); document.write( "Plug in n = 7
\n" ); document.write( "an = 2*(3)^(n-1)
\n" ); document.write( "a7 = 2*(3)^(7-1)
\n" ); document.write( "a7 = 1458
\n" ); document.write( "Repeat for n = 9
\n" ); document.write( "an = 2*(3)^(n-1)
\n" ); document.write( "a9 = 2*(3)^(9-1)
\n" ); document.write( "a9 = 13122
\n" ); document.write( "Both values match up, so we've confirmed this formula is correct.\r
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\n" ); document.write( "\n" ); document.write( "Now consider the case r = -3
\n" ); document.write( "an = a1*(r)^(n-1)
\n" ); document.write( "a7 = a1*(-3)^(7-1)
\n" ); document.write( "1458 = a1*(-3)^6
\n" ); document.write( "But hopefully you can see that (-3)^6 = 3^6 since the exponent is even.
\n" ); document.write( "Therefore, we'll land on the same starting a1 value we found earlier (a1 = 2).
\n" ); document.write( "We'll arrive at the formula an = 2*(-3)^(n-1)
\n" ); document.write( "I'll leave this check section to the student.\r
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\n" ); document.write( "\n" ); document.write( "We have two possibilities for the nth term formula:
\n" ); document.write( "an = 2*(3)^(n-1)
\n" ); document.write( "an = 2*(-3)^(n-1)\r
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\n" ); document.write( "\n" ); document.write( "Let's determine the 2nd term based on those formulas
\n" ); document.write( "an = 2*(3)^(n-1)
\n" ); document.write( "a2 = 2*(3)^(2-1)
\n" ); document.write( "a2 = 6
\n" ); document.write( "or
\n" ); document.write( "an = 2*(-3)^(n-1)
\n" ); document.write( "a2 = 2*(-3)^(2-1)
\n" ); document.write( "a2 = -6\r
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\n" ); document.write( "\n" ); document.write( "Conclusion:
\n" ); document.write( "The 2nd term is 6 if r = 3.
\n" ); document.write( "or
\n" ); document.write( "The 2nd term is -6 if r = -3.
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