91Ó°ÊÓ

Consider the sequence \(5,9,13,17,21, \ldots\) with \(a_{1}=5\) (a) Give a recursive definition for the sequence. (b) Give a closed formula for the \(n\) th term of the sequence. (c) Is 2013 a term in the sequence? Explain. (d) How many terms does the sequence \(5,9,13,17,21, \ldots, 533\) have? (e) Find the sum: \(5+9+13+17+21+\cdots+533\). Show your work. (f) Use what you found above to find \(b_{n},\) the \(n^{t h}\) term of \(1,6,15,28,45, \ldots,\) where \(b_{0}=1\)

Find the closed formula for each of the following sequences by relating them to a well known sequence. Assume the first term given is \(a_{1}\). (a) \(2,5,10,17,26, \ldots\) (b) \(0,2,5,9,14,20, \ldots\) (c) \(8,12,17,23,30, \ldots\) (d) \(1,5,23,119,719, \ldots\)

Use induction to prove for all \(n \in \mathbb{N}\) that \(\sum_{k=0}^{n} 2^{k}=2^{n+1}-1\).

Consider the sum \(4+11+18+25+\cdots+249\) (a) How many terms (summands) are in the sum? (b) Compute the sum using a technique discussed in this section.

Consider the sequence \(5,11,19,29,41,55, \ldots\) Assume \(a_{1}=5\) (a) Find a closed formula for \(a_{n},\) the \(n\) th term of the sequence, by writing each term as a sum of a sequence. Hint: first find \(a_{0},\) but ignore it when collapsing the sum. (b) Find a closed formula again, this time using either polynomial fitting or the characteristic root technique (whichever is appropriate). Show your work. (c) Find a closed formula once again, this time by recognizing the sequence as a modification to some well known sequence(s). Explain.

Consider the sequence \(\left(a_{n}\right)_{n \geq 1}\) that starts \(1,3,5,7,9, \ldots\) (i.e., the odd numbers in order). (a) Give a recursive definition and closed formula for the sequence. (b) Write out the sequence \(\left(b_{n}\right)_{n \geq 2}\) of partial sums of \(\left(a_{n}\right)\). Write down the recursive definition for \(\left(b_{n}\right)\) and guess at the closed formula.

Consider the sequence \(1,7,13,19, \ldots, 6 n+7\) (a) How many terms are there in the sequence? Your answer will be in terms of \(n\) (b) What is the second-to-last term? (c) Find the sum of all the terms in the sequence, in terms of \(n\).

Make up sequences that have (a) \(3,3,3,3, \ldots\) as its second differences. (b) \(1,2,3,4,5, \ldots\) as its third differences. (c) \(1,2,4,8,16, \ldots\) as its 100 th differences.

Consider the sequence \(1,3,7,13,21, \ldots\) Explain how you know the closed formula for the sequence will be quadratic. Then "guess" the correct formula by comparing this sequence to the squares \(1,4,9,16, \ldots\) (do not use polynomial fitting).

Consider the three sequences below. For each, find a recursive definition. How are these sequences related? (a) \(2,4,6,10,16,26,42, \ldots .\) (b) \(5,6,11,17,28,45,73, \ldots\) (c) \(0,0,0,0,0,0,0, \ldots\)