Telescopic Sum

[lemunsips]

Hello, I had a homework regarding telescopic sum (the teacher already went through it but I am still really confused).

The question was to find the explicit form of the sequence.

a₀ = 1, a_n+1 = a_n +n +1

The answer:
a0 = + 1
a₁ = a₀ + 0 + 1
a₂ = a₁ + 1 + 1
a₃ = a₂ + 2 + 1
a_n = a_n-1 +(n-1) + 1
Ʃⁿ _i=0 a_i = Ʃ^n-1 _i=0 a_i + Ʃ^n-1 _i=0 i + (n+1)(1)

[1]
[2]

(a₁ + a₂+…+a_n = a₁ + a₂ + … +a_n-1 )
a_n = Ʃ^n-1 _i=0 i + n + 1
[3]
a_n = (1+(n-1))(n-1) +n +1
= n(n-1)/2 + n+1
= ½ n² +1/2n +1

So after [1], I am not sure what is going on. [2] How did that super long equation suddenly turn into a_n? I know it has something to do with the cancellation of a₁ + a₂ +... in the brackets, but can someone help me clarify?
Also, after [3], im not sure how the previous equation from above jumped to that.

Sorry, i am kinda terrible at this topic and really trying to get used to that sigma symbol. Also, if you have any videos or examples that could further explain please share them.

Thank you!

 

[MarkFL]

I'm not seeing how we would get a telescoping series here, we are given the linear inhomogeneous difference equation:

\(\displaystyle a_{n+1}-a_{n}=n+1\) where \(\displaystyle a_0=1\)

The first thing we want to do is identify the homogeneous solution \(\displaystyle h_n\). Can you state the characteristic equation associated with this difference equation?

 

[lemunsips]

I'm not seeing how we would get a telescoping series here, we are given the linear inhomogeneous difference equation:

\(\displaystyle a_{n+1}-a_{n}=n+1\) where \(\displaystyle a_0=1\)

The first thing we want to do is identify the homogeneous solution \(\displaystyle h_n\). Can you state the characteristic equation associated with this difference equation?

Being genuinely honest, I have no idea what you are talking about (Sorry, i don't do math, I had no choice but to take this course).
Basically, I've been learning to convert recursive sequence into an explicit sequence. However, I can't simply apply it to the formula and make it into u_n = u₀ + nd

Apparently, I have to use a telescopic sum to solve it. It'll be easy to simply apply it, but I kinda want to understand what is going on in the solution I have presented just in case something similar pops up.