The Sum of the First 25 Multiples of 3: A Comprehensive Analysis
When determining the sum of the first 25 multiples of 3, one might approach the problem through various methods, each offering unique insights into the beauty of number theory. This article will explore those methods, including arithmetic progression, reversal techniques, and even a discussion on well-ordering integers.
Solution Using Arithmetic Progression
The sum of the first ( n ) terms of an arithmetic progression (AP) can be found using the formula:
[ S_n frac{n}{2} [2a (n - 1)d] ]For the first 25 multiples of 3, the first term ( a ) is 3, and the common difference ( d ) is also 3. Substituting these values into the formula, we get:
[ S_{25} frac{25}{2} [2 times 3 (25 - 1) times 3] ] [ S_{25} frac{25}{2} [6 72] ] [ S_{25} frac{25}{2} times 78 ] [ S_{25} 25 times 39 975 ]Solution Using Reversal Technique
Another approach involves adding the multiples of 3 in reverse order:
[ S 3 6 9 ldots 69 72 75 ] [ S 75 72 69 ldots 9 6 3 ]Adding vertically, we get:
[ 2S 78 78 78 ldots 78 78 78 ] (25 times) [ 2S 78 times 25 1950 ] [ S frac{1950}{2} 975 ]Solution Using Formula for Sum of an AP
The sum of the first 20 multiples of 3 can also be calculated using the formula for the sum of an AP:
[ S frac{n}{2} [a l] ]For the first 20 multiples of 3, the first term ( a ) is 3, and the last term ( l ) is 60. Substituting these values into the formula, we get:
[ S frac{20}{2} [3 60] ] [ S 10 times 63 630 ]Discussion on Well-Ordering of Integers
Another insightful approach involves the well-ordering of integers, where integers are ordered by their absolute value, and negative values precede positive values of the same absolute value. For multiples of 3, this ordering results in a pattern where every pair of positive and negative multiples of 3 cancels out. Therefore, the sum of the first 25 multiples of 3 (with 25 being odd) is 0.
For instance, the first multiples of 3 are 0, -3, 3, -6, 6, -9, 9, ..., and so on. After 0, the sum of each pair (0 and -3, 3 and -6, 6 and -9, etc.) cancels out to 0. With 25 being an odd number, this pattern holds true, resulting in a total sum of 0.
Understanding these different methods not only provides a deeper insight into the properties of arithmetic progressions and number theory but also demonstrates the elegance and versatility of mathematical problem-solving techniques.