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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

 Mar 20, 2021
 #1
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+2

(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! :)

 Mar 20, 2021

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