(1) What is the 50 th term of the arithmetic sequence with initial term 4 and common difference 3? 7 (2) Evaluate 4 X k=−3 (2k + 5). (nt: Since there are only eight terms in the sum, you can just write them all out and add.) (3) Evaluate 99 X i=0 ? − 2 3 ? i . (nt: Since there are 100 terms in the sum, it isn’t a good idea to write them all out and add. Use the formula for the sum of terms of a geometric sequence. Leave the answer with exponents rather than using a calculator to try to get a decimal approximation of the answer.) (4) (a) List the first four terms of the sequence defined recursively by a 0 = 2, and, for n ≥ 1, a n = 2a 2 n−1 − 1. (b) List the first five terms of the sequence with initial terms u 1 = 1 and u 2 = 5, and, for n ≥ 3, u n = 5u n−1 − 6u n−2 . Guess a closed form formula for the sequence. nt: The terms a
re simple combinations of powers of 2 and powers of 3. (5) Give a recursive definition of the geometric sequence with initial term a and common ratio r. nt: a n = ar n−1 isn’t a correct answer since this formula isn’t recursive. Make sure you write down a recursive formula: (1) give the initial term, and (2) give the rule for building new terms from previous terms. (6) (bonus) Express in summation notation: 1 2 + 1 4 + 1 6 + ··· + 1 2n , the sum of the reciprocals of the first n even positive integers. (Note that there are n terms in the sum.) nt: A common mistake on this question is using the symbol n both as an index for summation and to indicate the last term to be added in. To make sure you haven’t fallen into that trap, replace every n in your formula by a specific value, say 5. The result shouldbe a sum 1 2 + 1 4 + 1 6 + 1 8 + 1 10