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Sagot :
When evaluating the series we can identify that it is a arithmetic progression, this is because all the numbers are related to its previous number by the sum of the same ratio. We can see this below:
[tex]\begin{gathered} -171-(-175)=4 \\ -167-(-171)=4 \\ -163-(-167)=4 \end{gathered}[/tex]To calculate the sum of a series of this kind we can use the following expression:
[tex]S=\frac{n\cdot(a_1+a_n)}{2}[/tex]Where n is the number of terms the sequence has, in this case 67, a1 is the first term and an is the last term. To find the last term we can use the following expression:
[tex]a_n=a_1+(n-1)\cdot r_{}[/tex]Where r is the ratio for the series. Applying the data from the problem we have:
[tex]\begin{gathered} a_{67}=-175+(67-1)\cdot(4) \\ a_{67}=-175+264=89 \end{gathered}[/tex]Now we can calculate the sum for the sequence.
[tex]S=\frac{67\cdot(-175+89)}{2}=-2881[/tex]The series up to the 67th term has a value of -2881. The correct alternative is the last one.
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