What is the value of $\sum_{n=12}^{73} n$?
Answer (2)
Calculate the sum of integers from 1 to 73 using the formula: 2 73 ( 73 + 1 ) = 2701 .
Calculate the sum of integers from 1 to 11 using the formula: 2 11 ( 11 + 1 ) = 66 .
Subtract the second sum from the first to find the sum from 12 to 73: 2701 − 66 = 2635 .
The value of the sum is 2635 .
Explanation
Problem Analysis We are asked to find the sum of the integers from 12 to 73, inclusive. This can be written as:
Summation Notation n = 12 ∑ 73 n
Sum of Integers Formula We can calculate this sum by using the formula for the sum of an arithmetic series, or by finding the sum of the first 73 integers and subtracting the sum of the first 11 integers. The sum of the first n integers is given by 2 n ( n + 1 ) .
Sum from 1 to 73 The sum of the integers from 1 to 73 is:
Calculating the Sum n = 1 ∑ 73 n = 2 73 ( 73 + 1 ) = 2 73 × 74 = 73 × 37 = 2701
Sum from 1 to 11 The sum of the integers from 1 to 11 is:
Calculating the Sum n = 1 ∑ 11 n = 2 11 ( 11 + 1 ) = 2 11 × 12 = 11 × 6 = 66
Final Calculation Subtracting the sum of the integers from 1 to 11 from the sum of the integers from 1 to 73 gives the desired sum:
Result n = 12 ∑ 73 n = n = 1 ∑ 73 n − n = 1 ∑ 11 n = 2701 − 66 = 2635
Conclusion Therefore, the value of the sum is 2635.
Examples
Imagine you're stacking boxes, starting with 12 boxes on the bottom row, 13 on the next, and so on, up to 73 boxes on the top row. This problem helps you calculate the total number of boxes you've stacked. Understanding arithmetic series can guide you in scenarios such as calculating the total number of seats in a stadium with increasing rows, or determining the total amount of savings with consistent monthly deposits. This mathematical approach ensures efficient planning and resource management in practical tasks.
The sum of all integers from 12 to 73 can be calculated by first finding the sum from 1 to 73, which is 2701, and then subtracting the sum from 1 to 11, which is 66. This results in a total of 2635 for the sum from 12 to 73. Therefore, ∑ n = 12 73 n = 2635 .
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