document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1165192>Question 1165192</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Work out the next 5 terms in the number sequences below and explain your calculation using 'n' as your first term; e.g.n+1,n+2,n+3,etc:\r<BR>\n" );
document.write( "\n" );
document.write( "a. 297;290;276;255;.....\r<BR>\n" );
document.write( "\n" );
document.write( "b. 1;10;100;.....\r<BR>\n" );
document.write( "\n" );
document.write( "c. 3;9;81;.....\r<BR>\n" );
document.write( "\n" );
document.write( "d. 24;25;27;30;..... </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #853111 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2138)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=853111\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here are the next 5 terms for each number sequence, along with the explanation of the calculation using a term-to-term rule based on its position in the sequence, $n$.\r<BR>\n" );
document.write( "\n" );
document.write( "---\r<BR>\n" );
document.write( "\n" );
document.write( "## a. 297; 290; 276; 255; **227; 192; 150; 101; 45**\r<BR>\n" );
document.write( "\n" );
document.write( "### Analysis:<BR>\n" );
document.write( "This sequence is based on a decreasing difference that increases by a fixed amount each time (a quadratic sequence).\r<BR>\n" );
document.write( "\n" );
document.write( "| Term ($T_k$) | Value | Difference ($\Delta_1$) | Second Difference ($\Delta_2$) |<BR>\n" );
document.write( "| :---: | :---: | :---: | :---: |<BR>\n" );
document.write( "| $T_1$ | 297 | - | - |<BR>\n" );
document.write( "| $T_2$ | 290 | -7 | - |<BR>\n" );
document.write( "| $T_3$ | 276 | -14 | **-7** |<BR>\n" );
document.write( "| $T_4$ | 255 | -21 | **-7** |<BR>\n" );
document.write( "| $T_5$ | **227** | -28 | **-7** |<BR>\n" );
document.write( "| $T_6$ | **192** | -35 | **-7** |<BR>\n" );
document.write( "| $T_7$ | **150** | -42 | **-7** |<BR>\n" );
document.write( "| $T_8$ | **101** | -49 | **-7** |<BR>\n" );
document.write( "| $T_9$ | **45** | -56 | **-7** |\r<BR>\n" );
document.write( "\n" );
document.write( "### Next 5 Terms:<BR>\n" );
document.write( "227, 192, 150, 101, 45\r<BR>\n" );
document.write( "\n" );
document.write( "### Term-to-Term Rule:<BR>\n" );
document.write( "The difference decreases by 7 each time, starting with 7.<BR>\n" );
document.write( "* Next term ($n+1$): **$n - 28$**<BR>\n" );
document.write( "* $n+2$: **$(n-28) - 35$**<BR>\n" );
document.write( "* $n+3$: **$(n-28-35) - 42$**<BR>\n" );
document.write( "* ...and so on.\r<BR>\n" );
document.write( "\n" );
document.write( "---\r<BR>\n" );
document.write( "\n" );
document.write( "## b. 1; 10; 100; **1,000; 10,000; 100,000; 1,000,000; 10,000,000**\r<BR>\n" );
document.write( "\n" );
document.write( "### Analysis:<BR>\n" );
document.write( "This is a geometric sequence where each term is 10 times the previous term.\r<BR>\n" );
document.write( "\n" );
document.write( "### Next 5 Terms:<BR>\n" );
document.write( "1,000, 10,000, 100,000, 1,000,000, 10,000,000\r<BR>\n" );
document.write( "\n" );
document.write( "### Term-to-Term Rule:<BR>\n" );
document.write( "The next term is the previous term multiplied by 10.<BR>\n" );
document.write( "* Next term ($n+1$): **$n \times 10$**<BR>\n" );
document.write( "* $n+2$: **$(n \times 10) \times 10$**<BR>\n" );
document.write( "* $n+3$: **$(n \times 10 \times 10) \times 10$**<BR>\n" );
document.write( "* ...and so on.\r<BR>\n" );
document.write( "\n" );
document.write( "---\r<BR>\n" );
document.write( "\n" );
document.write( "## c. 3; 9; 81; **6,561; 43,046,721; $1.85 \times 10^{15}$; $3.40 \times 10^{30}$; $1.15 \times 10^{61}$**\r<BR>\n" );
document.write( "\n" );
document.write( "### Analysis:<BR>\n" );
document.write( "This sequence is generated by squaring the previous term.\r<BR>\n" );
document.write( "\n" );
document.write( "* $3^2 = 9$<BR>\n" );
document.write( "* $9^2 = 81$<BR>\n" );
document.write( "* $81^2 = 6,561$\r<BR>\n" );
document.write( "\n" );
document.write( "### Next 5 Terms:<BR>\n" );
document.write( "1.  $81^2 = \mathbf{6,561}$<BR>\n" );
document.write( "2.  $6,561^2 = \mathbf{43,046,721}$<BR>\n" );
document.write( "3.  $43,046,721^2 = \mathbf{1,853,020,188,851,841} \quad (\approx 1.85 \times 10^{15})$<BR>\n" );
document.write( "4.  $(\approx 1.85 \times 10^{15})^2 \approx \mathbf{3.43 \times 10^{30}}$<BR>\n" );
document.write( "5.  $(\approx 3.43 \times 10^{30})^2 \approx \mathbf{1.17 \times 10^{61}}$\r<BR>\n" );
document.write( "\n" );
document.write( "### Term-to-Term Rule:<BR>\n" );
document.write( "The next term is the square of the previous term.<BR>\n" );
document.write( "* Next term ($n+1$): **$n^2$**<BR>\n" );
document.write( "* $n+2$: **$(n^2)^2 = n^4$**<BR>\n" );
document.write( "* $n+3$: **$((n^2)^2)^2 = n^8$**<BR>\n" );
document.write( "* ...and so on.\r<BR>\n" );
document.write( "\n" );
document.write( "---\r<BR>\n" );
document.write( "\n" );
document.write( "## d. 24; 25; 27; 30; **34; 39; 45; 52; 60**\r<BR>\n" );
document.write( "\n" );
document.write( "### Analysis:<BR>\n" );
document.write( "This sequence is based on an increasing difference (an arithmetic progression of differences).\r<BR>\n" );
document.write( "\n" );
document.write( "| Term ($T_k$) | Value | Difference ($\Delta_1$) | Second Difference ($\Delta_2$) |<BR>\n" );
document.write( "| :---: | :---: | :---: | :---: |<BR>\n" );
document.write( "| $T_1$ | 24 | - | - |<BR>\n" );
document.write( "| $T_2$ | 25 | +1 | - |<BR>\n" );
document.write( "| $T_3$ | 27 | +2 | **+1** |<BR>\n" );
document.write( "| $T_4$ | 30 | +3 | **+1** |<BR>\n" );
document.write( "| $T_5$ | **34** | +4 | **+1** |<BR>\n" );
document.write( "| $T_6$ | **39** | +5 | **+1** |<BR>\n" );
document.write( "| $T_7$ | **45** | +6 | **+1** |<BR>\n" );
document.write( "| $T_8$ | **52** | +7 | **+1** |<BR>\n" );
document.write( "| $T_9$ | **60** | +8 | **+1** |\r<BR>\n" );
document.write( "\n" );
document.write( "### Next 5 Terms:<BR>\n" );
document.write( "34, 39, 45, 52, 60\r<BR>\n" );
document.write( "\n" );
document.write( "### Term-to-Term Rule:<BR>\n" );
document.write( "The difference increases by 1 each time, starting with 1.<BR>\n" );
document.write( "* Next term ($n+1$): **$n + 4$**<BR>\n" );
document.write( "* $n+2$: **$(n+4) + 5$**<BR>\n" );
document.write( "* $n+3$: **$(n+4+5) + 6$**<BR>\n" );
document.write( "* ...and so on. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );