Algebra
# Arithmetic Progressions

\[ 1 +2 + 3 + \cdots + 100. \]

What was Gauss' answer?

**Hint**: \(1 + 2 + 3 + \cdots + n = \frac 12 n(n+1) \).

An arithmetic sequence has 11 terms which sum to 220. What is the middle (6th) term in the sequence?

\[ 1 + 2 + 3 + \dots + (n-1) + n + (n-1) + \dots + 3 + 2 + 1 = 289 \]

In an arithmetic progression of 200 terms,

the 27th term equals to 2,

the 174th term equals to 4.

Find the sum of all the terms of the arithmetic progression.

\[\large 100+99+98+97+96+\cdots + 50 = \ ? \]

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