Sequence and series
t is an even number in the series 1/t +3/t +5/t +.... + t-1/t
(a)Determine the number of terms in the series in terms of t
(b) Determine the sum of the series in terms of t
+91 (a) The formula for the number of terms in an arithmetic sequence is:
(Last Term - The term before the first term)/ over the sequence increases by
Plugging this in, we get:
\({\frac {({\frac {t-1} {t}}-{\frac {-1} {t}})} {\frac {2} {t}}}={\frac {\frac {t} {t}} {\frac {2} {t}}}={\frac {1} {\frac {2} {t}}}={\frac {t} {2}}\)
Therefore there are t/2 terms in the sequence.
(b) The sum of an arithmetic sequence can be calculated with the formula:
Average of all terms * Number of terms.
The average of the terms in an arithmetic sequence can be found with the formula:
(First term + Last term)/2
Plugging this in, we get:
\({\frac {({\frac {1} {t}}+{\frac {t-1} {t}})} {2}={\frac {\frac {t} {t}} {2}}}={\frac {1} {2}}\)
Therefore the average of our terms is 1/2.
Finally, the sum of our terms is 1/2*t/2 =t/4
Solved! :)
