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# MTH 4150: Midterm 2 Practice 1

Problem 1

What is the coefficient next to $$x^4$$ in the power-series expansion of the function $$f(x)=(1+x+x^2+x^3+\cdots)^{10}$$?

Problem 2

Let $$S_n$$ be the number of ways to distribute $$n$$ identical apples to $$5$$ students such that the first two students have to get an even number of apples and the last three students have to get some number of apples divisible by $$3$$. Evaluate $$\sum_{n=0}^{\infty}\frac{S_n}{2^n}$$.

Problem 3

Find the generating function and the closed form for the sequence $$(a_n)_{n=0}^{\infty}$$ given by the following recurrence relation: $a_0=0,\;\;\; a_{n+1}=2a_n+n+5.$

Problem 4

Find the sequence $$(a_n)_{n=0}^{\infty}$$ whose exponential generating function is $H(X)=X+e^{3X}.$

Problem 5

Consider the sequence $$(a_n)_{n=0}^{\infty}$$ defined as $$a_n=n$$ for $$0\leq n\leq 5$$ and $$a_n=0$$ for $$n\geq 6$$. Let $$F$$ be the generating function for the sequence $$(a_n)_{n=0}^{\infty}$$. Determine $$F(2)$$.

Problem 6

Find the number of codewords of length $$5$$ from the alphabet $$\{a,b,c,d,e\}$$ in which $$b$$ occurs an odd number of times.

Problem 7

Let $$n$$ be a positive integer. Assume that: \begin{eqnarray*} N_k\mbox{ is the number of pairs }(a,b)\mbox{ of non-negative integers such that } ka+(k+1)b&=&n+1-k. \end{eqnarray*} Find $$N_1+N_2+\cdots+ N_{n+1}$$.