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Try this beautiful problem from HANOI 2018 based on Sequence and Series.

## Sequence and Series – HANOI 2018

Let {$u_n$} $n\geq1$ be given sequence satisfying the conditions $u_1=0$, $u_2=1$, $u_{n+1}=u_{n-1}+2n-1$ for $n\geq2$. find $u_{100}+u_{101}$

• is 13000
• is 10000
• is 840
• cannot be determined from the given information

### Key Concepts

Sequence

Series

Number Theory

But try the problem first…

Source

HANOI, 2018

Principles of Mathematical Analysis by Rudin

## Try with Hints

First hint

Here $u_2=1$, $u_3=3$, $u_4=6$, $u_5=10$

Second Hint

by induction $u_n=\frac{n(n-1)}{2}$ for every $n\geq1$

Final Step

Then $u_n+u_{n+1}$=$\frac{n(n-1)}{2}$+$\frac{n(n+1)}{2}$=$n^{2}$ for every $n\geq1$ Then $u_{100}+u_{101}$=10000.