# Question:Maple Unimodal

## Question:Maple Unimodal

Maple

Definition: A list or sequence of numbers
"a[1]"

"a[2]"

, ...,
"a[n]"

is unimodal if there is an index i such that

"a[1]"

<=
"a[2]"

<= . . . <=
"a[i]"

>=
"a[i + 1]"

>=
"a[i + 2]"

>= . . . >=
"a[n]"

That is, the sequence is non-decreasing up to some point after which it is non-increasing. Note that i can be 1 or n.  A constant sequence is considered to be unimodal.

Examples of unimodal lists:

[1, 1, 1, 1, 1],
[1,2,3,4,5,4,3,2,1],
[1, 2, 2, 3, 4, 5, 5, 5],
[5, 5, 4, 4, 3, 3, 1],
[1, 2, 2, 3, 3, 3, 4, 4, 2, 2, 1, 1, 1]

Examples of lists that are not unimodal:

[1, 0, 1, 0],
[1, 1, 2, 2, 3, 4, 5, 2, 2, 6, 4, 2, 2, 1, 0]

(a) Write a procedure unimodal to check whether or not a list of numbers is unimodal. The input should be a list and the output should be true or false.

One way to do this is to first write procedures called  increasing and decreasing which will check whether or not a sequence is non-decreasing (
"a[1]"

<=
"a[2]"

<=...) or non-increasing (
"a[1]"

>=
"a[2]"

>=...). Then for a list L use these procedures to check the two parts L[1..i] and L[i..n], n=nops(L) for each i. If you find an i such that the first list is "increasing" and the second is "decreasing" then you can return true.

I have up to here.

I need help with this part.

For n from 10 to 15  check whether or not the sequence of binomial coefficients

[binomial(n,0), binomial(n,1), binomial(n,2), . . ., binomial(n,n)]

is unimodal.

﻿