AMC 8 Daily Practice - Calculation Tricks by Grouping

Through observation, we notice that the expression appears repeatedly in the original formula.

The expression can be rewritten as:

Thus, the final result is .

We group the addends in pairs: We find that when calculating each of the expressions within the parentheses, the result always  is  one, and there are fifty parentheses. 

So the result of the original expression is .

By observing the arithmetic expression, we notice that the sum of the first four numbers is 0:

Examining the next set of four numbers, we find their sum is also 0:

Thus, the original expression can be rewritten as:

The problem now reduces to determining how many such groups exist. From 169 to 9, there are a total of 161 numbers.

When divided into groups of four, we get: groups with a remainder of 1.

Therefore, the original expression can be represented as:

The final result is .

By observing the arithmetic sequence, we notice that the sum of every two consecutive numbers from left to right is .

Therefore, we can group the numbers in pairs. Since there are numbers in total, dividing them into groups of two yields: groups with a remainder of .

Thus, the original expression can be rewritten as:

The final result is .

Through observation, we find that the sum of every two consecutive numbers is .

Grouping the sequence into pairs, we calculate the total number of terms as: numbers, which forms complete groups with 1 remaining term.

The original expression can thus be rewritten as: .

The final result is .

This problem is somewhat unique and requires careful observation. We notice that the operation signs repeat every three numbers in the pattern .

Therefore, we can group every three numbers together. The sum of the first group is , the sum of the second group is , and the sum of the third group is , indicating that these sums form an arithmetic sequence with a common difference of .

Calculating the number of groups: terms, then groups with a remainder of term. Thus, there are complete groups.

We can rewrite the original expression as:

This forms an arithmetic sequence with: - First term - Common difference - Last term -

Applying the arithmetic series sum formula (Gauss's formula):

The final result is .

Method 1:

By observation, we notice that the expression contains common factors 211 and 789.

Applying the associative property of multiplication, the original expression can be rewritten as:

Thus, the result is .

Method 2:

By observation, we notice that the expression contains common factors 555 and 445.

Applying the associative property of multiplication, the original expression can be rewritten as:

Thus, the result is .

By observing the first four terms, we notice the common factor .

Applying the associative property of multiplication, the first four terms can be rewritten as:

Similarly, analyzing terms five to eight, we identify the common factor and rewrite them as:

The original expression can thus be transformed into a series of such grouped terms:

This forms an arithmetic sequence with: - First term - Last term - Common difference - Number of terms

Applying Gauss's formula for the sum of an arithmetic series:

Final calculation:

The final result is .