Tocalculatethesumofallintegersfrom1to100,wecanusetheformulaforthesumofanarithmeticseries.Here'sadetailedbreakdown:

∪△∪ Step1:UnderstandtheProblem

˙ω˙ Weareaskedtofindthesumofallintegersfrom1to100,inclusive.Thisformsanarithmeticsequencewhere:

≥ω≤ Thefirstterm$a_1=1$

Thelastterm$a_n=100$

o(╯□╰)o Thecommondifference$d=1$

╯＾╰ Thenumberofterms$n=100$

≥▽≤ Step2:UsetheArithmeticSeriesSumFormula

Theformulaforthesum$S_n$ofthefirst$n$termsofanarithmeticsequenceis:

$$

S_n=\frac{n}{2}\times(a_1+a_n)

$$

ˇ▽ˇ Substitutetheknownvaluesintotheformula:

$n=100$

$a_1=1$

$a_n=100$

$$

(＊?↓˙＊) S_{100}=\frac{100}{2}\times(1+100)

$$

$$

˙▽˙ S_{100}=50\times101

$$

$$

˙﹏˙ S_{100}=5050

$$

￣□￣｜｜ Step3:VerificationviaPairingMethod

Anintuitivewaytoverifythisresultisbypairingterms:

$1+100=101$

$2+99=101$

≥▂≤ $3+98=101$

...

Thereare$\frac{100}{2}=50$suchpairs.

So,totalsum:

$$

(#｀′)凸 50\times101=5050

$$

＞▽＜ Thisconfirmsourcalculation.

FinalAnswer

$$

\boxed{5050}

$$