Hey there! As a supplier dealing with a wide range of products related to the number 3908032, I often get into some interesting number - related discussions. Today, I want to talk about a math problem: What is the sum of the first 3908032 even positive integers?
Let's start by understanding what an even positive integer is. An even positive integer can be represented in the form of (2n), where (n = 1,2,3,\cdots). The first even positive integer is (2\times1=2), the second is (2\times2 = 4), the third is (2\times3=6), and so on.
The sum (S_n) of the first (n) terms of an arithmetic series is given by the formula (S_n=\frac{n(a_1 + a_n)}{2}), where (n) is the number of terms, (a_1) is the first term, and (a_n) is the (n)th term.
For the series of even positive integers, (a_1 = 2). To find the (n)th term (a_n) of an arithmetic sequence, we use the formula (a_n=a_1+(n - 1)d), where (d) is the common difference. In the case of even positive integers, (d = 2).
So, when (n = 3908032), (a_1=2), and (d = 2). The (n)th term (a_{3908032}=a_1+(3908032 - 1)d). Substituting the values, we get (a_{3908032}=2+(3908032 - 1)\times2=2+3908031\times2=2\times(1 + 3908031)=2\times3908032).
Now, we use the sum formula (S_n=\frac{n(a_1 + a_n)}{2}). Substituting (n = 3908032), (a_1 = 2), and (a_{3908032}=2\times3908032) into the formula, we have:
(S_{3908032}=\frac{3908032\times(2 + 2\times3908032)}{2})
We can factor out the 2 from the numerator: (S_{3908032}=\frac{3908032\times2\times(1 + 3908032)}{2}).
The 2 in the numerator and denominator cancels out, and we get (S_{3908032}=3908032\times(3908033))
(3908032\times3908033=(3908000 + 32)\times(3908000+33))
Using the formula ((a + b)(a + c)=a^2+(b + c)a+bc), where (a = 3908000), (b = 32), and (c = 33)
(a^2=3908000^2=3908000\times3908000 = 15272464000000)
((b + c)a=(32 + 33)\times3908000=65\times3908000=254020000)
(bc=32\times33 = 1056)
(S_{3908032}=15272464000000+254020000 + 1056=15272718020000+1056=15272718021056)
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References:
- Arithmetic series formula: Basic knowledge of arithmetic series in mathematics textbooks
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