**Unformatted text preview: **n 2:: 2. 11. The sequence is 47,41, 35, 29, ... , which is arithmetic with a = 47 andd = -6. So S = ~[2a+ (n-l)d] = 50[94 + 99( -6)] = -25000. 12. This is the sum of an arithmetic sequence with a = 529, d = -4 and an = -459. Since an = a + (n -l)d, we obtain n = 498. The sum is n 498 S = 2[2a + (n -1)d] = 2[1058 - 4(497] = 249( -930) = -231,570. l3. For n = I, the formula gives ~(1 -(_!)O) = ~(1 -1) = 0 = al. For n = 2, the formula gives ~(1 -(_!)l) = ~(1 + !) = l = a2. Thus the result is true for n = 1 and n = 2. Now assume that k > 2 and that the formula is correct for all n < k. We wish to prove that the result is true when n = k. We have ak = !(ak-2 + ak-l) (given) = ! [~(1 -(_!)k-3) + ~(1 _ (_!)k-2)] (the induction hypothesis) _ ! [~ _ ~(_!)k-3 + ~ _ ~ (_!)k-2] -29 9 2 9 9 2 = H~) [2 -(_!)k-3(1 -!)] = H~) [2 + (_!)k-3(_!)] = ~(1-(_!)k-l)...

View
Full
Document