Final answer:
To find the sum from 1 to 547, we can pair the numbers so that each pair sums to 548, then multiply the number of pairs by this sum and add the median number. Using this method by Gauss, the total sum is 149,998.
Step-by-step explanation:
To find the sum of a sequence from 1 to 547 without using a calculator or a formula, we can use the method made famous by the German mathematician Carl Friedrich Gauss. The trick involves pairing the numbers in the sequence so that each pair adds up to the same total.
First, we notice that 1 + 547 = 548, and 2 + 546 = 548 as well, and this pattern continues. There are 547 numbers in total, so we can make 547/2 = 273.5 pairs. Since we cannot have half a pair, it means there is one number in the middle of the sequence (the median) which does not form a pair. This median number is 274 (since 273 numbers before and 273 numbers after). So, we have 273 pairs each adding up to 548.
Now to find the total sum, we multiply the pairs by the common sum of each pair, and add the median number:
273 pairs * 548 per pair + 274 = 149,724 + 274 = 149,998.
This method shows that by applying simple arithmetic operations and logical reasoning, we can calculate large sums quite efficiently. It illustrates how by being approximate or applying shortcuts, we can often find solutions more quickly.