Professor Thesis is puzzling over the formula for the present value of a stream of payments of $1 a year, starting 1 year from now and continuing forever. He knows that the value of this stream is expressed by the infinite series

but he can€™t remember the simplified formula for this sum. All he knows is that if the first payment were to arrive today, rather than a year from now, the present value of the sum would be $1 higher. So he knows that

Professor Antithesis suffers from a similar memory lapse. He can€™t remember the formula for S either. But, he knows that the present value of $1 a year forever, starting right now has to be 1 + r times as large as the present value of $1 a year, starting a year from now. (This is true because if you advance any income stream by a year, you multiply its present value by 1+r.) That is,

(a) If Professor Thesis and Professor Antithesis put their knowledge together, they can express a simple equation involving only the variable S. This equation is S +1 _______. Solving this equation, they find that ________.
(b) The two professors have also forgotten the formula for the present value of a stream of $1 per year starting next year and continuing for K years. They agree to call this number S (K) and they see that

Professor Thesis notices that if each of the payments came 1 year earlier, the present value of the resulting stream of payments would be

Professor Antithesis points out that speeding up any stream of payments by a year is also equivalent to multiplying its present value by (1 + r). Putting their two observations together, the two professors noticed an equation that could be solved for S (K). This equation is S (K) + 1 €“ 1 / (1+r) K = _______ Solving this equation for S (K), they find that the formula for S (K) is ___________.

S(K) = (1+1) + (1t, . . . + (1+r)R =S(K) +1-(1+r)K K-1