document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=569553>Question 569553</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Peter works as an electrical engineer for an electrical firm,His total salary for his first year of work at this firm is $60,000.peter's annual salary increases by 10% of his first year's salary,Find\r<BR>\n" );
document.write( "\n" );
document.write( "I)His total salary after 10 years<BR>\n" );
document.write( "II) the least value of 'n' fir which his total salary after 'n' years is more than 2milion dollars\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "b) John is an electrical engineer in a rival firm,His total salary for the first year is also $60,000.But his annual salary follows a geometric progession.If he earns $75,600 in his 4th year of work,find\r<BR>\n" );
document.write( "\n" );
document.write( "i)his annual percentage increase in salary,<BR>\n" );
document.write( "II) his total salary after 10 years </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #367576 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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/cgi-bin/show-face.mpl?user=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=367576\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I believe they mean that poor Peter, gets a raise of (0.10)($60,000)= $6,000 every year. His salary goes up in an arithmetic sequence (or arithmetic progression).<BR>\n" );
document.write( "In the 10th year, after 9 raises, he gets $60,000+9($6,000)=$60,000+$54,000=$114,000.<BR>\n" );
document.write( "The sum of an arithmetic progression is the average of first and last terms, multiplied times the number of terms.<BR>\n" );
document.write( "I) Adding up Peter’s 10 years of salary we get 10($60,000+$114,000)/2=$870,000. I believe That’s what the problem means by “His total salary after 10 years.”<BR>\n" );
document.write( "II) Another way to calculate the sum <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5Bn%5D ALIGN=MIDDLE  ALT=\"S%5Bn%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the first <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > terms of an arithmetic progression starting with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D ALIGN=MIDDLE  ALT=\"a%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > and increasing by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d ALIGN=MIDDLE  ALT=\"d\"   DISABLED_style= \"max-width:90%; height:auto;\" > from term to term is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5Bn%5D=%28n%2F2%29%282+a%5B1%5D%2B%28n-1%29d%29 ALIGN=MIDDLE  ALT=\"S%5Bn%5D=%28n%2F2%29%282+a%5B1%5D%2B%28n-1%29d%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "In Peter’s case, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D=60000 ALIGN=MIDDLE  ALT=\"a%5B1%5D=60000\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=6000 ALIGN=MIDDLE  ALT=\"d=6000\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5Bn%5D=%28n%2F2%29%282+%2A60000%2B%28n-1%296000%29+=%28n%2F2%29%28120000%2B6000n-6000%29=%28n%2F2%29%28114000%2B6000n%29=57000n%2B3000n%5E2 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "To find out when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5Bn%5D%3E=2000000 ALIGN=MIDDLE  ALT=\"S%5Bn%5D%3E=2000000\"   DISABLED_style= \"max-width:90%; height:auto;\" > we solve<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=57000n%2B3000n%5E2=2000000 ALIGN=MIDDLE  ALT=\"57000n%2B3000n%5E2=2000000\"   DISABLED_style= \"max-width:90%; height:auto;\" > ---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=57n%2B3n%5E2-2000=0 ALIGN=MIDDLE  ALT=\"57n%2B3n%5E2-2000=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3n%5E2%2B57n-2000=0 ALIGN=MIDDLE  ALT=\"3n%5E2%2B57n-2000=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=%28-57+%2B-+sqrt%28+57%5E2-4%2A3%2A%28-2000%29+%29%29%2F%282%2A3%29+ ALIGN=MIDDLE  ALT=\"n=%28-57+%2B-+sqrt%28+57%5E2-4%2A3%2A%28-2000%29+%29%29%2F%282%2A3%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > using the quadratic formula. We simplify to<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=%28-57+%2B-+sqrt%28+3249%2B24000%29%29%2F6=%28-57+%2B-+sqrt%2827249%29%29%2F6 ALIGN=MIDDLE  ALT=\"n=%28-57+%2B-+sqrt%28+3249%2B24000%29%29%2F6=%28-57+%2B-+sqrt%2827249%29%29%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" > , which rounds to 18.01<BR>\n" );
document.write( "That means that 18 years is not quite enough. In fact<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5B18%5D=57000%2A18%2B3000%2A18%5E2=1998000 ALIGN=MIDDLE  ALT=\"S%5B18%5D=57000%2A18%2B3000%2A18%5E2=1998000\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "Peter will have collected $2,000,000 in salary at some point during his 19th year, and after 19 years he will have totaled $2,166,000 in earnings, so I suspect the answer expected for part II) is 19.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5B19%5D=57000%2A19%2B3000%2A19%5E2=2166000 ALIGN=MIDDLE  ALT=\"S%5B19%5D=57000%2A19%2B3000%2A19%5E2=2166000\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "b)John’s annual salary follows a geometric progression starting at $60,000, and earns $75,600 in his 4th year of work. <BR>\n" );
document.write( "I) Let’s find his annual percentage increase in salary,<BR>\n" );
document.write( "In a geometric sequence (or geometric progression), each term is the product of the term before times a common ratio, r. After raises at the end of the first 3 years, John’s salary for the fourth year is<BR>\n" );
document.write( "$60,000<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5E3 ALIGN=MIDDLE  ALT=\"r%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >=$75,000, so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5E3 ALIGN=MIDDLE  ALT=\"r%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >=$75,000/$60,000=5/4=1.25 and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=root%283%2C1.25%29 ALIGN=MIDDLE  ALT=\"r=root%283%2C1.25%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >= 1.0772 (rounded)<BR>\n" );
document.write( "Each year end he gets a raise worth 0.0772 times the prior year salary, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%287.72%29 ALIGN=MIDDLE  ALT=\"highlight%287.72%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >% of the prior year salary.<BR>\n" );
document.write( "John may look at Peter in envy at the end of the first year, but his raise is always 7.72% of the latest salary, and keeps growing, while Peter’s raises are always the same $6,000 (10% of his starting salary).  Peter had a salary of $60,000+3($6,000) =$78,000 for the fourth year, but soon John overtakes Peter in salary<BR>\n" );
document.write( "II) Let’s find John’s total salary after 10 years<BR>\n" );
document.write( "For the 10th year, John’s annual salary should be $60,000<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%285%2F4%29%5E%281%2F3%29%29%5E9 ALIGN=MIDDLE  ALT=\"%28%285%2F4%29%5E%281%2F3%29%29%5E9\"   DISABLED_style= \"max-width:90%; height:auto;\" >=$60,000<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285%2F4%29%5E3 ALIGN=MIDDLE  ALT=\"%285%2F4%29%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >=$117,187.50 (already ahead of Peter)<BR>\n" );
document.write( "However, he is not yet ahead in terms of total earnings.<BR>\n" );
document.write( "The sum of n terms of a geometric progression of common ratio <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r ALIGN=MIDDLE  ALT=\"r\"   DISABLED_style= \"max-width:90%; height:auto;\" >, starting with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D ALIGN=MIDDLE  ALT=\"a%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > is calculated as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D%28r%5En-1%29%2F%28r-1%29 ALIGN=MIDDLE  ALT=\"a%5B1%5D%28r%5En-1%29%2F%28r-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "In this case $60,000<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%281.077217%5E10-1%29%2F%281.077217-1%29%29 ALIGN=MIDDLE  ALT=\"%28%281.077217%5E10-1%29%2F%281.077217-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >= $60,000<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%282.10394-1%29%2F%281.077217-1%29%29 ALIGN=MIDDLE  ALT=\"%28%282.10394-1%29%2F%281.077217-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >= $60,000<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281.10394%2F0.077217%29 ALIGN=MIDDLE  ALT=\"%281.10394%2F0.077217%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >= $857,791.83 (rounded) </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );