Search problems

35 problems match “sequences”. Clear

Counting alternating-parity increasing sequences ●●●○○ 50 ptsA finite strictly increasing sequence of integers is called alternating if its first term is odd, its second term is even, its third term is odd, and so on, alternating in parity. The empty sequence also counts as alternating. Let $A(n)$ be the number of alternating sequences whose terms are all taken from $\{1,2,\ldots,n\}$ , so $A(1)=2$ and $A(2)=3$ . Find $A(20)$ .
Longest edge of a rectangular block ●●●○○ 50 ptsThe lengths of three unequal edges of a rectangular solid block are in geometric progression. The volume of the block is $216\ \text{cm}^3$ and its total surface area is $252\ \text{cm}^2$ . Find the length of the longest edge, in cm.
Complex sequence with conjugate recurrence ●●●○○ 50 ptsThe complex number sequence $\{z_n\}$ satisfies $z_1=1$ and $z_{n+1}=\overline{z_n}+1+ni$ for $n=1,2,3,\dots$ , where $i$ is the imaginary unit and $\overline{z_n}$ is the conjugate of $z_n$ . Find the real part and the imaginary part of $z_{2015}$ .
Counting case-switching letter sequences ●●●●○ 100 ptsBob writes a sequence of letters of the alphabet, each of which is either uppercase or lowercase, under these rules. If he has just written an uppercase letter, he either writes the same letter in lowercase next, or the next letter of the alphabet in uppercase. If he has just written a lowercase letter, he either writes the same letter in uppercase next, or the preceding letter of the alphabet in lowercase. For example, $a\,A\,a\,A\,B\,C\,D\,d\,c\,b\,B\,C$ is such a sequence. How many sequences of 32 letters can he write that start at lowercase $a$ and end at lowercase $z$ ?
Possible values of the largest removed integer ●●●●○ 100 ptsYann writes down the first $n$ consecutive positive integers, $1,2,3,4,\ldots,n-1,n$ . He removes four different integers $p,q,r,s$ from the list. At least three of $p,q,r,s$ are consecutive and $100<p<q<r<s$ . The average of the integers remaining in the list is $89.5625$ . What is the number of possible values of $s$ ?