For any integerk,  let αk=coskπ7+isinkπ7, where i=−1.The value of the expression ∑k=112αk+1−αk∑k=13α4k−1−a4k−2 is

We have   αk=coskπ7+isinkπ7=cos2kπ14+isin2kπ14=ei2kπ14⇒αk+1=ei2k+1π14∴αk+1−αk=ei2k+1π14−ei2kπ14=ei2kπ14ei2π14−1=eiπ7−1   ∵eiθ=1Also we have α4k−1−α4k−2=eiπ7−1Now ∑k=112αk+1−αk∑k=13α4k−1−a4k−2=∑k=112eiπ7-1∑k=13eiπ7-1=123=4