document.write( "Question 847870: What is the formula for this sequence:
\n" ); document.write( " 21, 45, 77, 117, 165 ?
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Algebra.Com's Answer #510717 by swincher4391(1107)\"\" \"About 
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45 -21 = 24
\n" ); document.write( "77-45 = 32
\n" ); document.write( "117 - 77 = 40...
\n" ); document.write( "etc. It seems as though our difference is increasing by 8 each time.
\n" ); document.write( "By knowing that the 2nd difference is 8, we know that the function is of the form 4n^2 + something. In general, an^2 has second difference 2a.
\n" ); document.write( "We also realize that the difference between a(0) and a(1) is 16.
\n" ); document.write( "So a(0)= 5.
\n" ); document.write( "So we have 4n^2 + b*n + 5
\n" ); document.write( "This should be enough to figure out what b is.
\n" ); document.write( "Take a_2 for instance.
\n" ); document.write( "4(2)^2 + b*2 + 5 = 45
\n" ); document.write( "16 + 2b + 5 = 45
\n" ); document.write( "2b = 24
\n" ); document.write( "b = 12
\n" ); document.write( "So our entire function a(n) = 4n^2 + 12n + 5 <---- ANSWER
\n" ); document.write( "We check our answer:
\n" ); document.write( "a(1) = 4 + 12 + 5 = 21
\n" ); document.write( "a(2) = 4*4 + 24 + 5 = 45
\n" ); document.write( "a(3) = 4*9 + 36 + 5 = 77
\n" ); document.write( "a(4) = 4*16 + 48 + 5 = 117
\n" ); document.write( "a(5) = 4*25 + 60 + 5 = 165
\n" ); document.write( "Just for fun a(10) = 4*100 + 12*10 + 5 = 525. Simple. \n" ); document.write( "
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