sec summation techniques once againwe still do not have proof that (nn( )/ whybecause we did not prove that ( )/ ( )/ nor that ( )/ ( ( )( we merely hypothesized patterns from looking at few terms fortunatelyit is easy to check our hypothesis with induction example solve the summation / = we will begin by writing out table of the first few values of the summation and see if we can detect pattern ( (ng(nf ( by direct inspection of the second line of the tablewe might recognize the pattern ( - simple induction proof can then prove that this always holds true alternativelyconsider if we hadn' noticed the pattern for the form of (nwe might observe that (nappears to be reaching an asymptote at one in which casewe might consider looking at the difference between (nand the expected asymptoteby subtracting (nfrom ( the result is shown in the last line of the tablewhich have clear pattern of / from this we can easily deduce guess that ( againa simple induction proof will verify the guess example solve the summation (nn ari ar ar arn = this is called geometric series our goal is to find some variation for (nsuch that subtracting one from the other leaves us with an easily manipulated equation because the difference between consecutive terms of