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Topic
First Poster
Last Poster
.
Let 
.
. Prove that
2019 PMO Qualifying Stage III.3
wonderboy807 2
N
by aops-g5-gethsemanea2
A sequence \{a_n\}_{n \geq 1} of positive integers is defined by a_1 = 2 and for integers n > 1,
\frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_{n-1}} + \frac{n}{a_n} = 1.
Determine the value of \sum_{k=1}^{\infty} \frac{3^k(k+3)}{4^k a_k}.
\frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_{n-1}} + \frac{n}{a_n} = 1.
Determine the value of \sum_{k=1}^{\infty} \frac{3^k(k+3)}{4^k a_k}.
2 replies
a tst 2013 test
Math2030 6
N
by Math2030
Given the sequence 
and 
. Prove that
not is a perfect square for all 
.
and 
. Prove that
not is a perfect square for all 
.
6 replies
MTG MOSTP 2025 Handout Problem.
wonderboy807 2
N
by LilKirb
Let S_n = \sqrt{1 + \frac{1}{1^2} + \frac{1}{2^2}} + \sqrt{1 + \frac{1}{2^2} + \frac{1}{3^2}} + ... + \sqrt{1 + \frac{1}{n^2} + \frac{1}{(n+1)^2}} What is the numerical ratio of \frac{S_{2024}}{2024}?
2 replies
Original Problem: Geometry and Functions
wonderboy807 1
N
by wonderboy807
For any positive integer n, let F(n) be the number of interior diagonals in a convex polygon with n+3 sides. Find 1/(F(1)) + 1/(F(2)) + ... + 1/F(10))
Answer:
Answer:
1 reply
[Sipnayan SHS] Written Round, Average, #4.6
LilKirb 4
N
by Shan3t
Define the function
![\[f(n) = \sum_{k=0}^{n} \binom{n}{k} a_k, \quad n = 1, 2, 3, \ldots\]](//latex.artofproblemsolving.com/9/3/7/9377324cb6585daaf2cbec7fed6fce2209835e22.png)
where
![\[a_k =
\begin{cases}
3^k, & \text{if } k \text{ is even}, \\
0, & \text{if } k \text{ is odd}.
\end{cases}
\]](//latex.artofproblemsolving.com/8/b/e/8be520a30bdfd9272163470ee69120c2fd3e9b59.png)
Find the remainder when 
is divided by 
![\[f(n) = \sum_{k=0}^{n} \binom{n}{k} a_k, \quad n = 1, 2, 3, \ldots\]](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 
4 replies
Remaining balls
pacoga 4
N
by mafj
A bag contains 
white balls, 
red balls and 
black balls. We extract balls at random without replacement until the bag becomes empty. What is the probability that there are still black and red balls in the bag when the last white ball is drawn?
white balls, 
red balls and 
black balls. We extract balls at random without replacement until the bag becomes empty. What is the probability that there are still black and red balls in the bag when the last white ball is drawn?4 replies
Linear algebra
Feynmann123 4
N
by OGMATH
Hi everyone,
I was wondering whether when I tried to compute e^(2x2 matrix) and got the expansions of sinx and cosx with the method of discounting the constant junk whether it plays any significance. I am a UK student and none of this is in my School syllabus so I was just wondering…
I was wondering whether when I tried to compute e^(2x2 matrix) and got the expansions of sinx and cosx with the method of discounting the constant junk whether it plays any significance. I am a UK student and none of this is in my School syllabus so I was just wondering…
4 replies
