document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1198920>Question 1198920</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I'm saving money over a year in order to buy a $157.50 gift. I start with an empty piggy bank. In January I put in a small amount of money. Each month after January, I put in $1.75 more than I put in the previous month. By the end of December, I've saved $157.50 exactly. The amount in dollars that I had in my piggy bank after April's deposit was </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #832592 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=832592\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "x = amount deposited in January\r<BR>\n" );
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document.write( "Since the deposits increase by the same amount ($1.75) each month, we have an arithmetic sequence\r<BR>\n" );
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document.write( "a1 = first term = x<BR>\n" );
document.write( "d = common difference = 1.75<BR>\n" );
document.write( "n = number of terms\r<BR>\n" );
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document.write( "We'll have n = 12 to represent the 12 months.<BR>\n" );
document.write( "Let's find the sum of the first 12 terms of this sequence.<BR>\n" );
document.write( "This will get the total amount saved after December's deposit.\r<BR>\n" );
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document.write( "Sn = sum of the first n terms of an arithmetic sequence<BR>\n" );
document.write( "Sn = (n/2)*(2*a1+d*(n-1))<BR>\n" );
document.write( "S12 = (12/2)*(2*x+1.75*(12-1))<BR>\n" );
document.write( "S12 = 12x+115.5\r<BR>\n" );
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document.write( "The sum of the first 12 terms of the sequence {x, x+1.75, (x+1.75)+1.75, etc} is 12x+115.5\r<BR>\n" );
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document.write( "Set this equal to 157.50 which is the total amount saved at the end of December.\r<BR>\n" );
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document.write( "Solve for x.<BR>\n" );
document.write( "12x+115.5 = 157.50<BR>\n" );
document.write( "12x = 157.50-115.5<BR>\n" );
document.write( "12x = 42<BR>\n" );
document.write( "x = 42/12<BR>\n" );
document.write( "x = 3.5\r<BR>\n" );
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document.write( "You deposited $3.50 into the piggybank in January to start things off.<BR>\n" );
document.write( "Then for February, the deposit is 3.50+1.75 = 5.25 and so on until you arrive at December.\r<BR>\n" );
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document.write( "I don't recommend writing out all the terms as the tutor @greenestamps has done, if doing it by hand. <BR>\n" );
document.write( "Though you can use computer software to do it for you. A spreadsheet is really handy in situations like this.\r<BR>\n" );
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document.write( "Let's return back to the summation formula mentioned earlier.<BR>\n" );
document.write( "This time we plug in n = 4 to determine how much is saved after April's deposit.<BR>\n" );
document.write( "I.e. this is the sum of the first four terms.\r<BR>\n" );
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document.write( "Sn = (n/2)*(2*a1+d*(n-1))<BR>\n" );
document.write( "S4 = (4/2)*(2*3.5+1.75*(4-1))<BR>\n" );
document.write( "S4 = 24.5\r<BR>\n" );
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document.write( "Answer: <font color=red>$24.50</font><BR>\n" );
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