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Sum Of First N Integers

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Subject
Mathematics
Type
Homework
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So let’s get started…
This problem we are trying to solve is called “The sum of the first n positive integers
So let’s assume we have n integers lined up to be summed.
Mathematically, we can formulate the sum as follows:
𝑺 = 𝟏 + 𝟐 + 𝟑 + 𝟒 + 𝟓 + 𝟔 + + 𝒏 Equation (1)
Where S represents the sum of the first n numbers, well we don’t know how much S is but it’s a
start.
Now let’s do a simple trick:
𝑺 = 𝒏 +
(
𝒏 − 𝟏
)
+
(
𝒏 − 𝟐
)
+
(
𝒏 𝟑
)
+
(
𝒏 − 𝟒
)
+
(
𝒏 𝟓
)
+ + 𝟏 Equation (2)
Don’t worry, all I did is start summing from the right to left.
Basically the following two sums are the same aren’t they? Both sum to 21.
1 + 2 + 3 + 4 + 5 + 6 = 6 + 5 + 4 + 3 + 2 + 1
On the same token Equation (1) and Equation (2) are telling the same thing.
Thus, let’s move on and try to find the mathematical expression for S (the sum of the firs n positive
integers)
If I try to sum Equation (1) and Equation (2), I would get the following result:
𝑺 = 𝟏 + 𝟐 + 𝟑 + 𝟒 + 𝟓 + 𝟔 + + 𝒏
𝑺 = 𝒏 +
(
𝒏 − 𝟏
)
+
(
𝒏 − 𝟐
)
+
(
𝒏 𝟑
)
+
(
𝒏 − 𝟒
)
+
(
𝒏 𝟓
)
+ + 𝟏
𝟐𝑺 = (𝒏 + 𝟏) +
(
𝒏 − 𝟏
)
+ (𝒏 + 𝟏) + (𝒏 + 𝟏) + (𝒏 + 𝟏) + (𝒏 + 𝟏) + ⋯+ (𝒏 + 𝟏)
Summing the terms on the left side of the equal sign gives us 2S (S +S), and on the right side of the
equal sign we sum the first term of Equation (1) with the corresponding first term of Equation (2)
and so on
1 + n = (n+1); 2 + (n - 1) = (n + 1)
Thus we have always (n + 1) when we sum up the terms on the right side of the equal sign.
Remember that we have n terms on the right side of the equal sign, meaning having (n +1) terms n
times can be simplified as follows:

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So let’s get started… This problem we are trying to solve is called “The sum of the first n positive integers” So let’s assume we have n integers lined up to be summed. Mathematically, we can formulate the sum as follows: 𝑺 = 𝟏 + 𝟐 + 𝟑 + 𝟒 + 𝟓 + 𝟔 + ⋯+ 𝒏 Equation (1) Where S represents the sum of the first n numbers, well we don’t know how much S is but it’s a start. Now let’s do a simple trick: 𝑺 = 𝒏 + (𝒏 − 𝟏) + (𝒏 − 𝟐) + (𝒏 − 𝟑) + (𝒏 − 𝟒) + (𝒏 − 𝟓) + ⋯ + 𝟏 Equation (2) Don’t worry, all I did is ...
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