What is the Sum of the First 10 Terms of the AP 15 10 5?

The sum of the first 10 terms of the arithmetic progression (AP) 15, 10, 5 is -75.

Understanding Arithmetic Progressions (AP)

An arithmetic progression is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the common difference (d).

• First term (a): The first number in the sequence. In this case, a = 15.
• Common difference (d): The constant difference between consecutive terms. Here, d = 10 - 15 = -5.
• Number of terms (n): The number of terms in the sequence we want to sum. In this question, n = 10.

Calculating the Sum of an AP

The sum (S_n) of the first n terms of an arithmetic progression can be calculated using 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.

Applying the Formula to the Given AP

Let's plug in the values from the AP 15, 10, 5 into the formula:

• a = 15
• d = -5
• n = 10

S₁₀ = (10/2) [2(15) + (10-1)(-5)]
S₁₀ = 5 [30 + (9)(-5)]
S₁₀ = 5 [30 - 45]
S₁₀ = 5 [-15]
S₁₀ = -75

Therefore, the sum of the first 10 terms of the AP 15, 10, 5 is -75, as confirmed by the reference.