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Sum Of First N Positive Integers

The sum of the first n positive integers is the total obtained by adding all natural numbers from 1 up to a specific number n. In the context of arithmetic progressions, this sequence forms an AP with a first term of 1 and a common difference of 1.

Key Concepts

The sequence of positive integers 1, 2, 3, up to n forms an Arithmetic Progression where the first term a is 1 and the common difference d is 1.

The sum can be derived using the general AP sum formula Sn = n/2 [2a + (n-1)d] by substituting a = 1 and d = 1.

It can also be derived using the first and last term formula Sn = n/2 (a + l), where the first term a is 1 and the last term l is n.

Formula & Equation

Sn = n(n + 1) / 2

Sn represents the sum of the first n positive integers, and n represents the total number of terms or the last integer in the sequence.

Common Misconceptions

Myth: Students often apply the formula n(n+1)/2 to find the sum of any sequence of numbers.

Fact: This specific formula only works for consecutive positive integers starting exactly from 1. If the sequence starts from a different number or skips numbers, the general AP sum formula must be used.

Real World Applications

Calculating the total number of objects arranged in a triangular grid, such as a stack of cans in a grocery store.

Determining the total number of handshakes in a room if everyone shakes hands with each other exactly once.

Frequently Asked Questions

How do you find the sum of the first 100 positive integers?

Using the formula Sn = n(n + 1) / 2, substitute n with 100 to get 100 multiplied by 101 divided by 2, which equals 5050.

Is zero included in the sum of the first n positive integers?

No, zero is neither positive nor negative. The sequence of positive integers strictly begins with the number 1.

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