Algebra
# Arithmetic Progressions

As a young boy, Gauss was able to calculate math quickly. He was asked for the value of

$1 +2 + 3 + \cdots + 100.$

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?

$n$ in the following expression?

What is the value of$1 + 2 + 3 + \dots + (n-1) + n + (n-1) + \dots + 3 + 2 + 1 = 289$

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.

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