Which expression is equivalent to $\sum_{n=1}^{35} n$ ?
A. $\frac{35(35+1)}{2}$
B. $\frac{35(35+1)(2 \cdot 35+1)}{6}$
C. $\left[\frac{35(35+1)}{2}\right]^2$
D. $\frac{35(35-1)}{2}$
Answer (2)
Recall the formula for the sum of the first n natural numbers: ∑ i = 1 n i = 2 n ( n + 1 ) .
Substitute n = 35 into the formula: 2 35 ( 35 + 1 ) .
The expression equivalent to ∑ n = 1 35 n is 2 35 ( 35 + 1 ) .
The final answer is: 2 35 ( 35 + 1 )
Explanation
Understanding the Problem We are asked to find an expression equivalent to the sum of the first 35 natural numbers. This is a classic problem in mathematics that can be solved using a well-known formula.
Recalling the Formula 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 = 35 . So we need to substitute this value into the formula.
Applying the Formula Substituting n = 35 into the formula, we get: 2 35 ( 35 + 1 ) This is the expression we are looking for.
Comparing with Options Now, let's compare this result with the given options:
2 35 ( 35 + 1 )
6 35 ( 35 + 1 ) ( 2 ⋅ 35 + 1 )
[ 2 35 ( 35 + 1 ) ] 2
2 35 ( 35 − 1 )
The first option, 2 35 ( 35 + 1 ) , matches our result.
Final Answer Therefore, the expression equivalent to ∑ n = 1 35 n is 2 35 ( 35 + 1 ) .
Examples
The sum of natural numbers has many practical applications. For example, if you are arranging seats in a theater where each row has one more seat than the previous row, calculating the total number of seats involves summing natural numbers. Similarly, in computer science, understanding the summation of natural numbers is crucial for analyzing the time complexity of algorithms, especially those involving loops. Knowing how to quickly compute these sums allows for efficient planning and resource allocation in various scenarios.
The expression equivalent to ∑ n = 1 35 n is 2 35 ( 35 + 1 ) , which corresponds to Option A. This formula accurately computes the sum of the first 35 natural numbers. Thus, the chosen answer is Option A.
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