Sum of an Arithmetic Progression: -2 -5 -8… -227

Let's explore the sum of the arithmetic progression (AP) where the first few terms are -2, -5, -8, and so on, up to -227. This article will guide you through the process step-by-step and explain the underlying formulas.

Understanding the Sequence

The given sequence is an arithmetic progression (AP) with a common difference of -3. An arithmetic progression is a sequence of numbers such that the difference of any two successive members is a constant, which in this case is -3.

Determining the Number of Terms (n)

To find the sum of all the terms in this AP, we need to determine the number of terms (n). We are given the first term (a) and the last term (an) of the sequence:

a (first term): -2 d (common difference): -3 an (last term): -227

The nth term of an arithmetic progression can be found using the formula:

"an a (n-1)d"

Substitute the known values into the formula:

-227 -2 (n-1)(-3)

Simplify and solve for n:

-227 -2 - 3n 3 -227 1 - 3n -228 -3n n 76

Hence, there are 76 terms in this AP.

Calculating the Sum (Sn)

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

"Sn (n/2) [2a (n-1)d]"

Substitute the known values into the formula:

"S76 (76/2) [2(-2) (76-1)(-3)]"

Calculate step-by-step:

"S76 38 [-4 75(-3)]" "S76 38 [-4 - 225]" "S76 38 (-229)" "S76 -8702"

Therefore, the sum of all the terms in the AP -2, -5, -8, ..., -227 is -8702.

Conclusion

This method can be used to find the sum of any arithmetic progression by determining the number of terms and applying the sum formula.