Sum of the First n Natural Numbers

Transcript

Sum of the First n Natural Numbers Natural numbers are numbers starting from 1, like 1, 2, 3, 4, 5, …. Let’s find sum of first 10 natural numbers Thus, S = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 Now, to find S, we write our numbers from largest to smallest S = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 S = 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 Adding both equations S + S = 11 + 11 + 11….. 10 times 2S = 11 × 10 S = 𝟏/𝟐 × 11 × 10 S = 11 × 5 S = 55 Now, let’s look at first Sum of first n natural numbers Thus, S = 1 + 2 + 3 + 4 + 5 + …. + n And, writing it form largest to smallest as well S = 1 + 2 + 3 + 4 + 5 + …. + n S = n + (n – 1) + (n – 2) + (n – 3) + (n – 4) + … + 1 Thus, 2S = (n + 1) + (n + 1) + (n + 1) …. _ … N times 2S = (n + 1) × n S = 1/2 × (n + 1) × n S = 𝟏/𝟐 𝒏(𝒏+𝟏) Thus, our formula becomes Sum of first n natural numbers = 𝟏/𝟐 𝒏(𝒏+𝟏) S + S = 11 + 11 + 11….. 10 times 2S = 11 × 10 S = 𝟏/𝟐 × 11 × 10 S = 11 × 5 S = 55 Now, let’s look at first Sum of first n natural numbers Thus, S = 1 + 2 + 3 + 4 + 5 + …. + n And, writing it form largest to smallest as well S = 1 + 2 + 3 + 4 + 5 + …. + n S = n + (n – 1) + (n – 2) + (n – 3) + (n – 4) + … + 1 Thus, 2S = (n + 1) + (n + 1) + (n + 1) …. _ … N times 2S = (n + 1) × n S = 1/2 × (n + 1) × n S = 𝟏/𝟐 𝒏(𝒏+𝟏) Thus, our formula becomes Sum of first n natural numbers = 𝟏/𝟐 𝒏(𝒏+𝟏) Thus, 2S = (n + 1) + (n + 1) + (n + 1) …. _ … N times 2S = (n + 1) × n S = 1/2 × (n + 1) × n S = 𝟏/𝟐 𝒏(𝒏+𝟏) Thus, our formula becomes Sum of first n natural numbers = 𝐒_𝐧=𝟏/𝟐 𝒏(𝒏+𝟏)