Calculate the sum of the first 15 positive even numbers. - Coaching Toolbox
Calculate the Sum of the First 15 Positive Even Numbers: A Simple & Effective Guide
Calculate the Sum of the First 15 Positive Even Numbers: A Simple & Effective Guide
When learning basic arithmetic and number patterns, one common exercise is calculating the sum of the first n positive even numbers. Whether you're a student, teacher, or math enthusiast, understanding how to compute this efficiently can save time and boost mathematical confidence. In this article, we explore how to calculate the sum of the first 15 positive even numbers step by step, using both manual calculation and shortcut formulas.
What Are the First 15 Positive Even Numbers?
Understanding the Context
Positive even numbers begin from 2 and increase by 2 each time. The sequence starts:
2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30
There are 15 terms in this sequence, all divisible by 2, and following the pattern:
2 × 1, 2 × 2, 2 × 3, ..., 2 × 15
Why Does This Matter?
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Key Insights
Knowing how to sum arithmetic sequences is valuable in mathematics and computer science. It helps lay the foundation for topics like series, summation formulas, and weighted sums. This particular problem is also great practice for mental math and pattern recognition.
Method 1: Adding Them Manually
A straightforward way to calculate the sum is to add each number from 2 to 30 (only the even ones) sequentially. While simple, this method becomes tedious for larger n. For the first 15 even numbers:
2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20 + 22 + 24 + 26 + 28 + 30
Add step-by-step:
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- (2 + 30) = 32
- (4 + 28) = 32
- (6 + 26) = 32
- (8 + 24) = 32
- (10 + 22) = 32
- (12 + 20) = 32
- (14 + 18) = 32
- 16 (the middle term)
There are 7 pairs of 32, and one leftover 16:
7 × 32 = 224
224 + 16 = 240
✅ Sum of the first 15 positive even numbers is 240.
Method 2: Using the Formula for the Sum of an Arithmetic Series
There’s a quick, efficient formula for summing the first n even numbers:
Sum = n × (first term + last term) ÷ 2
For positive even numbers:
- First term (a₁) = 2
- Last term (aₙ) = 2n
- Number of terms (n) = 15
Plug in values:
Sum = 15 × (2 + 30) ÷ 2
Sum = 15 × 32 ÷ 2
Sum = 15 × 16
Sum = 240
