To find the sum of the first 40 positive integers (i.e., 1 + 2 + 3 + 4 + … up to 40), a arithmetic progression

Arithmetic progression-Quick Calculation: The Sum of First 40 Positive Numbers

What is A.P?

An Arithmetic Progression (AP) is a sequence of numbers arranged in a specific order, where the difference between any two consecutive terms remains constant. In other words, each successive number in the sequence is obtained by adding a fixed value, called the common difference, to the previous number.

For example, in the sequence 1, 3, 5, 7, 9, and so on, the difference between each term is consistently 2.

Moreover, this constant difference can be either positive or negative, which means the sequence can either increase or decrease, depending on the value of the common difference.

Additionally, if the common difference is zero, all terms in the sequence will be the same.

Therefore, an arithmetic progression is defined not only by its starting value but also by its common difference, making it a predictable and structured sequence.

The general formula for the sum of the first n terms of an arithmetic progression is:

S_n = (n/2) (2a + (n-1)d)

Where:

• Sₙ is the sum of the first n terms
• n is the number of terms
• a is the first term
• d is the common difference between the terms

Given values:

• a (the first term) = 1
• d (the common difference) = 1
• n (the number of terms) = 40

Hence Substituting the given values into the formula:

S40 = (40/2)(2(1) + (40 – 1) 1)

Simplifying step by step:

hence calculate :

1. ( (40/2)= 20 )
2. ( 2(1) = 2 )
3. ( 40 – 1 = 39 )
4. Multiply ( 39 × 1 = 39 )

Hence substitute these into the expression:

S₄₀= 20 (2 + 39)

= 20 ×41

Finally, S₄₀= 820

Thus, the sum of the first 40 positive integers is 820.

More examples Unlocking the secrets of Arithmetic series: A complete guide: