Let A(x) be the series A(x) lowing sequences in terms of compositions and sums involving the expression A(x) (rather than En>o (something). Here is an example: En>o anx". Express the generating series for each of the fol- 0, the nth term of the sequence is Sequence: — ао, а1, —а2, аз, —а4, а5, .... (–1)"+lan.) (Starting with n = Solution: The generating series for this sequence is given by: E(-1)"+'a anx" n>0 n20 = -En(-2)* n>0 = -A(-x) (а) 0, ао, 0, а1,0, а2, .... (Starting with n = 0, the nth term is 0 if n is even, and a: if n is odd.) (The 0th term is ao, and for all n > 1 the nth term is given (b) ао, ал + 2ао, а2 + 2ал, аз + 2а2, .... by an + 2an-1.)

you get stuck on a question, I will be happy to give ou a hint. 1. Write out the first 5 terms (starting with ao) of each of the sequences described below. Then give either 'a closed formula or a recursive definition for the sequence (whichever is NOT given in the problem). (a) an = }(n² +n). (b) an = 2an-1 – an-2 With ao = (c) an = na,n-1 with ao = 1. %3D 0 and a1 = 1. %3D %3D 2. For each sequonco giuon hel

Write the first four terms of the sequence, where n = 1, 2, 3, 4, ... n3 an = 1+ n2 (A) 글, 몽, 유, 몸 1 8 27 64 (B) §, 3 7' 5 1 8 27 64 (D) 2: 5· 10’ 17 . . 12 3 6 2' 5 10' 17 3 6 9 12 (C) ž, 3, 4' 5

Select a formula for the general term of the sequence, assuming that the pattern of the first few terms continues. 2 4 8 } 7' 49' 343 Oan (²)^ (²)*** Oºn O.-(-1)^ = Oªn Oº, · (²)*** = = (²)^ O₁-(-3)*** Oºn = (-³)^¯¹ O₁₁- (-²)^ = 1