document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1203542>Question 1203542</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The first term of an AP is 7 and the sum of the first 40 terms is 1216.<BR>\n" );
document.write( "find the nth term and 20th term </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #839193 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=839193\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> here's a reference.<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://byjus.com/maths/arithmetic-progression/\" target = \"_blalnk\">https://byjus.com/maths/arithmetic-progression/</a>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the formulas are:<BR>\n" );
document.write( "an = a + (n-1) * d<BR>\n" );
document.write( "sn = n/2 * (2a + (n-1) * d)<BR>\n" );
document.write( "sn = n/2 * (a + l)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "an is the nth term<BR>\n" );
document.write( "a is the first term, also able to be called a1.<BR>\n" );
document.write( "n is the number of terms in the progression.<BR>\n" );
document.write( "d is the common difference between the value of a term and the value of the term that comes next in the progression.<BR>\n" );
document.write( "sn is the nth term in the progression.<BR>\n" );
document.write( "l is the last term in the progression.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "as example, let n = 5 and d = 2 and a = 3\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "a5 = a + (n-1) * 2 becomes a5 = 3 + 4 * 2 which becomes 11.<BR>\n" );
document.write( "the progression will be 3,5,7,9,11, confirming the formula is dorrect.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "s5 = n/2 * (2a + (n-1) * d) becomes s5 = 5/2 * (2 * 3 + 4 * 2) which becomes 5/2 * (6 + 8) which becomes 5/2 * 14 which becomes 35.<BR>\n" );
document.write( "the sum of the progression is 3 + 5 + 7 + 9 + 11 which is equal to 35, confirming the formula is correct.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the other formula for sn is s5 = n/2 * (a + l) which becomes 5/2 * (3 + 11) which becomes 5/2 * 14 which becomes 35, confirming the formula is correct.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "in your problem, you have:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "a = 7<BR>\n" );
document.write( "s40 = 1216\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you want to find an and a20.<BR>\n" );
document.write( "n is equal to 40, as given in the description of the problem.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "start with the sn formula, which is sn = n/2 * (2a + (n-1) * d)<BR>\n" );
document.write( "since n = 40 and a = 7 and sn = 1216, the formula becomes 1216 = 20 * (14 + 39 * d)<BR>\n" );
document.write( "simplify to get 1216 = 280 + 780 * d<BR>\n" );
document.write( "subtract 280 from both sides of the equation to get 936 = 780 * d<BR>\n" );
document.write( "solve for d to get d = 936 / 780 = 1.2\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you now have:\r<BR>\n" );
document.write( "\n" );
document.write( "a = 7<BR>\n" );
document.write( "n = 40<BR>\n" );
document.write( "d = 1.2<BR>\n" );
document.write( "sn = 1216\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you can now solve for an.<BR>\n" );
document.write( "formula is an = a + (n-1) * d which becomes a40 = 7 + 39 * 1.2 which becomes a40 = 53.8.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "your alternate formula for sn is sn = n/2 * (a + l) which becomes sn = 40/2 * (7 + 53.8) which becomes sn = 20 * 60.8 which becomes 1216.<BR>\n" );
document.write( "since this is the same as given, this confirms that value of l is 53.8 and the formula of sn = n/2 * (a + l) is correct.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "finally, you are asked to the find the value of the 20th term.<BR>\n" );
document.write( "an = a + (n-1) * d becomes a20 = 7 + 19 * 1.2 which becomes 29.8.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "as a bonus, the sum of the first 20 terms will use the formula of s20 = 20/2 * (7 + 29.8) which is equal to 368.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "i used excel to confirm all these values are correct.<BR>\n" );
document.write( "here's the excel worksheet display.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2023/090901.jpg\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );