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# Arithmetic Progressions

What's the sum of the first 100 positive integers? How about the first 1000? Learn the fun and fast way to solve problems like this.

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

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

What was Gauss' answer?

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

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Victor Paes Plinio**

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

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Mohamed Ahmed Abd El-F...**

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What is the value of \( n \) in the following expression?

###### Image Credit: Ricardo Liberato, CC SA 2.0

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

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Arron Kau**

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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.

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Somesh Singh**

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

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Dharma Teja**

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