Question 1202468
<pre>
Calculate the sum between and including the terms t16 and t53 of an arithmetic sequence with first term 543 and common difference  -  12.

use the A.P. formula to get the value of term 16 (t<sub>16</sub> or a<sub>16</sub>).
So, we get: {{{matrix(1,3, a[n], "=", a[1] + (n - 1)d)}}} ---- Formula for a specific term of an A.P./A.S.
           {{{matrix(4,3, a[16], "=", 543 + (16 - 1)- 12, a[16], "=", 543 - 12(15), a[16], "=", 543 - 180, a[16], "=", 363)}}}

Now, we use the formula for the sum of an A.P. series from term 16 to term 53. The number of terms (n) in this series = 38 (53 - 16 + 1)
Additionally, note that since we're summing the values from the 16th term to the 53rd term, we will use term 16 as the 1st term of the series.

So, we get: {{{matrix(1,3, S[n], "=", (n/2)(2a[1] + (n - 1)d))}}} ---- Formula for the sum of an A.P./A.S.
           {{{matrix(1,3, S[38], "=", (38/2)(2(363) + (38 - 1)- 12))}}} ---- Substituting 38 for n, 363 for a<sub>1</sub>, and - 12 for d
           {{{highlight_green(matrix(2,3, S[38], "=", (38/2)(726 - 12(37)), S[38], "=", 19(726 - 444)))}}}

<font size  = 4><font color = red><b>From term 16 (t<sub>16</sub>, or a<sub>16</sub>) to term 53 (t<sub>53</sub>, or a<sub>53</sub>), SUM of the series, or S<sub>38</sub> = 19(282) = 5,358</font></font></b>

<font size  = 4><font color = blue><b>OR</font></font></b>

Find the sum of ALL 53 terms:
            {{{matrix(1,3, S[n], "=", (n/2)(2a[1] + (n - 1)d))}}} ---- Formula for the sum of an A.P./A.S.
           {{{matrix(1,3, S[53], "=", (53/2)(2(543) + (53 - 1)- 12))}}} ---- Substituting 53 for n, 543 for a<sub>1</sub>, and - 12 for d
           {{{highlight_green(matrix(4,3, S[53], "=", (53/2)("1,086" - 12(52)), S[53], "=", (53/2)("1,086" - 624), S[53], "=", (53/2)(462), S[53], "=", (53/2)2(231)))}}}

SUM of ALL 53 terms = 53(231) = 12,243

Find the sum of the FIRST 15 terms:
            {{{matrix(1,3, S[n], "=", (n/2)(2a[1] + (n - 1)d))}}} ---- Formula for the sum of an A.P./A.S.
           {{{matrix(1,3, S[15], "=", (15/2)(2(543) + (15 - 1)- 12))}}} ---- Substituting 15 for n, 543 for a<sub>1</sub>, and - 12 for d
           {{{highlight_green(matrix(4,3, S[15], "=", (15/2)("1,086" - 12(14)), S[15], "=", (15/2)("1,086" - 168), S[15], "=", (15/2)(918), S[15], "=", (15/2)2(459)))}}}

SUM of FIRST 15 terms = 15(459) = 6,885

Now, SUBTRACT the SUM of the 1st 15 terms from the SUM of ALL 53 terms to get the SUM of terms 16 to 53.
This is: <font size  = 4><font color = red><b>12,243 - 6,885 = 5,358</font></font></b></pre>