What is the sum of the first 200 odd natural numbers?
[tex]S_{200} =[/tex]
Answer (2)
Represent the sum of the first 200 odd natural numbers as S 200 = ∑ n = 1 200 ( 2 n − 1 ) .
Rewrite the sum using linearity: S 200 = 2 ∑ n = 1 200 n − ∑ n = 1 200 1 .
Apply the formula for the sum of the first n natural numbers and the sum of a constant: S 200 = 2 ( 2 200 ( 201 ) ) − 200 .
Calculate the final result: S 200 = 40200 − 200 = 40000 .
Explanation
Problem Analysis We are asked to find the sum of the first 200 odd natural numbers. Let's denote this sum as S 200 .
Expressing the Sum The n -th odd natural number can be represented as 2 n − 1 . Therefore, the sum of the first 200 odd natural numbers can be written as: S 200 = n = 1 ∑ 200 ( 2 n − 1 )
Splitting the Sum We can use the properties of summation to split this into two separate sums: S 200 = 2 n = 1 ∑ 200 n − n = 1 ∑ 200 1
Sum of First n Natural Numbers We know that the sum of the first n natural numbers is given by the formula: i = 1 ∑ n i = 2 n ( n + 1 ) In our case, n = 200 , so: n = 1 ∑ 200 n = 2 200 ( 200 + 1 ) = 2 200 × 201 = 100 × 201 = 20100
Sum of 1 repeated n times The sum of 1, 200 times, is simply 200: n = 1 ∑ 200 1 = 200
Calculating the Final Sum Now, substitute these results back into the expression for S 200 :
S 200 = 2 ( 20100 ) − 200 = 40200 − 200 = 40000
Final Answer Therefore, the sum of the first 200 odd natural numbers is 40000.
Alternatively, we can use the formula for the sum of the first n odd natural numbers, which is n 2 . In this case, n = 200 , so the sum is 20 0 2 = 40000 .
Examples
Understanding the sum of odd numbers has practical applications in various fields. For instance, consider tiling a square area with smaller square tiles. If you start with a single tile (1x1), then add a layer of 3 tiles to make a 2x2 square, then add a layer of 5 tiles to make a 3x3 square, and so on, the total number of tiles you've used at each stage is the sum of consecutive odd numbers. Knowing that the sum of the first n odd numbers is n 2 allows you to quickly calculate the number of tiles needed to form a larger square without counting each tile individually. This concept is useful in construction, design, and even in certain computer algorithms.
The sum of the first 200 odd natural numbers is 40000 . This can be calculated using the formula for the sum of the first n odd numbers, n 2 , where n = 200 . Alternatively, it can also be computed using summation notation and properties of summation, yielding the same result.
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