Question 1195430

The first term of an AP is 3. Given that the sum of the first 6 terms is 48 and that the sum of all the terms is 168 .calculate the common difference,the number of terms in the AP and the last term 
<pre>                                          Sum of an AP: {{{matrix(1,3, S[n], "=", (n/2)(2a[1] + (n - 1)d))}}}
Sum of this AP with its first 6 terms summing to 48: {{{matrix(5,3, S[6], "=", (6/2)(2(3) + (6 - 1)d), 48, "=", 3(6 + 5d), 3(16), "=", 3(6 + 5d), 16, "=", 6 + 5d, 10, "=", 5d)}}}
                                   Common difference, or {{{highlight_green(matrix(1,5, d, "=", 10/5, "=", highlight(2)))}}}

                                          Sum of an AP: {{{matrix(1,3, S[n], "=", (n/2)(2a[1] + (n - 1)d))}}}
       Sum of this AP with its terms summing to 168: {{{matrix(6,3, S[n], "=", (n/2)(2(3) + (n - 1)2), 168, "=", (n/2)(6 + 2n - 2), 168, "=", (n/2)(4 + 2n), 168, "=", (n/2)2(2 + n), 168, "=", 2n + n^2, n^2 + 2n - 168, "=", 0)}}}
                                                 (n - 12)(n + 14) = 0
            <font color = red><font size = 4><b> Number of terms in series</font></font></b>, or <font color = red><font size = 4><b> n = 12</font></font></b>  or <font color = red><font size = 4><b> n = - 14 (ignore)</font></font></b> 

                                       Specific term of an AP: {{{matrix(1,3, a[n], "=", a[1] + (n - 1)d)}}}
                                 Last term (12<sup>th</sup>) of this AP: {{{matrix(2,3, a[12], "=", 3 + (12 - 1)2, a[12], "=", 3 + 22)}}}
                      <font color = red><font size = 4><b> Last term (12<sup>th</sup>) of this AP</font></font></b>, or <font color = red><font size = 4><b> a<sub>12</sub> = 25</font></font></b></pre>