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2. A function f is defined for all positive integers and satisfiesf(1)= 1996,f(1) +f(2) +... +f(n)=n?f(n)for all n > 1.Calculate the exact value of f(1996).
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(1)If you have never seen anything like this before, it seems reasonable tostart calculating:f(1)=1996(given);f(2)=(from f(1)+f(2)=2’f(2))f(3)=_,and so on. Use this approach to find f(2), f(3), and f(4).(2)If you persevere, and keep your wits about you, you might just noticesomething interesting (though you must be careful not to let the nambers1996, 1, 2, 3, 4, etc. obscure what is going on). lf you are lucky, you may evenbe able to guess a value for f(1996). But this would not answer the question!In mathematics, the word 'determine’means more than just 'guess’or 'find’it means that you have to show exactly why your value is correct. In otherwords, you have to 'find the correct value, and prove it is correct’. For this, it isnot the values of f(2), f(3),and so on, that matter, but their form. Thus it isimportant to express f(2) and f(3) in a form that reveals what is really goingon:
f(2)=(22_1'/(1),r)-3' u+r@l-g'rot a'-n ol(2-1)(32-Df(1).Write out the calculation which shows that
3222f(4)=(22-1)(32-1)'(42-1)f(1).
(3)Now guess what you expect to be the corresponding expression for f(n)interms of f(1), and prove that your guess is correct (by induction on n).(4)Even at this stage it is important to resist the temptation simply tosubstitute n=1996. Factorize each ofthe factors(r2-1) in the denominatorof your (proven!) expression for f(n), and cancel to obtain a greatly simplifiedformula for f(n)in terms of n and f(1).Finally, substitute n=1996.