chap analysis techniques the summation is factor of rwe can shift terms if we multiply the entire expression by rrf (nr ari ar ar ar arn+ = we can now subtract the one equation from the otheras followsf (nrf (na ar ar ar arn (ar ar ar arn arn+ the result leaves only the end termsf (nrf (nn ari = ari = ( ) (na ar + thuswe get the result (na arn+ - where example for our second example of the shifting methodwe solve (ni = we can achieve our goal if we multiply by twon ( ( + = the ith term of (nis + while the ( )th term of (nis ( + subtracting one expression from the other yields the summation of and few non-canceled termsn (nf ( = = = = + shift ' value in the second summationsubstituting ( for