Question 1202468
.
Calculate the sum between and including the terms t16 and t53 of an arithmetic sequence 
with first term 543 and common difference -12.
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            The solution by @mananth is a pile of mistakes,

            that do not deserve to analyse them.


            I came to bring a correct solution.



<pre>
There are two ways to solve the problem.


                          First way


The needed sum is the difference  S(t16,t53) = S(53) - S(15).      (1)



The sum of the first 53 terms is (use the standard formula for the sum
of n first terms of an AP)

    S(53) = {{{(a[1] + ((n-1)*d)/2)*n}}} = {{{(543 + 52*(-12)/2)*53}}} = 12243.     (2)



The sum of the first 15 terms is (use the same standard formula for the sum
of n first terms of an AP)

    S(15) = {{{(a[1] + ((n-1)*d)/2)*n}}} = {{{(543 + 14*(-12)/2)*15}}} = 6885.      (3)


Now the  <U>ANSWER</U>  is the difference of numbers (2) and (3), according to (1):

    S(t16,t53) = S(53) - S(15) = 12243 - 6885 = 5358.             (4)    <U>ANSWER</U>



                          Second way


Second way is to calculate the terms of this AP

    t16 = 543 + (16-1)*(-12) = 363,  t53 = 543 + (53-1)*(-12) = -81,


and then use another standard formula for the sum 

    S(t16,t53) = {{{((t16+t53)/2)*(53-16+1)}}} = {{{((363 - 81)/2)*38}}} = 5358.    (5)


We get the same number/answer as in (4).


<U>ANSWER</U>.  The needed sum is 5358,  calculated in two different ways.
</pre>

Solved.



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Regarding work of @mananth at this forum, I always repeat it many times,
that his work can be successful under two indispensable conditions:


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;- two persons should assist him: one to explain him what to do, 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;and the second to re-write and edit his compositions after him.