What is the Sum of the First 20 Terms of the Arithmetic Progression 5, 9, 13 if the Common Difference is 4?

The sum of the first 20 terms of the arithmetic progression 5, 9, 13, with a common difference of 4, is 860.

Here's a breakdown of how to find that sum and how the reference verifies this result:

Understanding Arithmetic Progressions

An arithmetic progression (AP) is a sequence of numbers where the difference between any two consecutive terms is constant. This constant difference is called the common difference (d). In this case, d = 4. The first term (a) in this sequence is 5.

Formula for Sum of an Arithmetic Progression

The sum (S_n) of the first 'n' terms of an arithmetic progression is given by the formula:

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

Where:

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

Calculation

In our specific problem:

• n = 20
• a = 5
• d = 4

Let's substitute these values into the formula:
S₂₀ = (20/2) [2(5) + (20 - 1)4]
S₂₀ = 10 [10 + (19)4]
S₂₀ = 10 [10 + 76]
S₂₀ = 10 86
S₂₀ = 860

Verification using provided reference:

The provided reference states:
*∴ The required sum = 860

This confirms that the sum of the first 20 terms is indeed 860, as calculated using the formula.