Question 1140702
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The series is arithmetic, with a common difference of 2/t.<br>
(1) To find the number of terms in an arithmetic series...
find the difference between the last and first terms;
divide by the common difference; and
add 1.<br>
{{{(t-1)/t-1/t = (t-2)/t}}}
{{{((t-2)/t)/(1/t) = t-2}}}
{{{(t-2)/2 = t/2-1}}}
{{{(t/2-1)+1 = t/2}}}<br>
ANSWER (1): The number of terms in the series is t/2.<br>
(2) The sum of the terms of any series is the number of terms, multiplied by the average of all the terms; since this sequence is arithmetic, that means the sum is the number of terms, multiplied by the average of the first and last terms.<br>
{{{(t/2)*((1/t)+(t-1)/t)/2}}}
{{{(t/2)*((t/t)/2)}}}
{{{(t/2)*(1/2) = t/4}}}<br>
ANSWER: (2): The sum of the terms in the series is t/4.<br>
(3) The terms in this series are all of the form in the above discussion.  The values of t in the terms of this series are 4, 6, 8, ..., and 50.  So the sum of this series, using the result in (2), is<br>
{{{4/4+6/4+8/4}}} ... + {{{50/4}}}<br>
This is again an arithmetic series; the sum is number of terms, multiplied by the average of the first and last:<br>
{{{24((4/4+50/4)/2) = 24(54/8) = 54*3 = 162}}}