is a nonconstant arithmetic sequence such that 
and the terms 
form a geometric sequence in that order. Find the smallest index 
for which 
.
Prove that
Let 
Prove that
.
be positive reals such that 
,
of 
.
![\[f(n) = \sum_{k=0}^{n} \binom{n}{k} a_k, \quad n = 1, 2, 3, \ldots\]](http://latex.artofproblemsolving.com/9/3/7/9377324cb6585daaf2cbec7fed6fce2209835e22.png)
where![\[a_k =
\begin{cases}
3^k, & \text{if } k \text{ is even}, \\
0, & \text{if } k \text{ is odd}.
\end{cases}
\]](http://latex.artofproblemsolving.com/8/b/e/8be520a30bdfd9272163470ee69120c2fd3e9b59.png)
Find the remainder when 
is divided by 
distinct penguin chicks, 
are selected to form the Penchick Manga Club. Then, one of these 
be the number of ways to select the club and assign its president. Compute 
.
is 
where 
are relatively prime positive integers, find 
.
Y by
Summary of the problem: If I try to post anything with PHP (a coding language), it
Page URL: In any forum or private messages
Steps to reproduce:
1. Create a post.
2. Put some PHP inside, can't give example
Expected behavior: Should post the message
Frequency: 100%
Operating system(s): macOS Sequoia 15.2.1
Browser(s), including version: Safari
Additional information: See attachments
Page URL: In any forum or private messages
Steps to reproduce:
1. Create a post.
2. Put some PHP inside, can't give example
Expected behavior: Should post the message
Frequency: 100%
Operating system(s): macOS Sequoia 15.2.1
Browser(s), including version: Safari
Additional information: See attachments
Attachments:
