What is the formula to find the sum of the first n terms of an AP?

The formula to find the sum of the first n terms of an Arithmetic Progression (AP) is:

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

Where:

• S_n represents the sum of the first n terms.
• n is the number of terms you are summing.
• a is the first term in the Arithmetic Progression.
• d is the common difference between consecutive terms.

Explanation of the Formula

The formula is derived from the average of the first and last term, multiplied by the number of terms. This can be expressed as:

S_n = n * (a + l)/2

Where l is the last term. Since l can also be written as a + (n-1)d, substituting this into the equation gives us:

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

Example

Let's say we want to find the sum of the first 10 terms of an AP where the first term (a) is 2 and the common difference (d) is 3. Using the formula:

S₁₀ = 10/2 [2(2) + (10-1)3]
S₁₀ = 5 [4 + (9)3]
S₁₀ = 5 [4 + 27]
S₁₀ = 5 [31]
S₁₀ = 155

Therefore, the sum of the first 10 terms is 155.