AMC 8 Daily Practice - Consecutive Reduction

We observe that this problem contains repeated numbers and division symbols, so we expand the parentheses and rewrite the original expression as:

We notice that consecutive pairs cancel out completely.

Therefore, the final result of the original expression is .

The problem involves mixed numbers.

We first convert each mixed number to an improper fraction:

Final result:

First, we calculate the value inside each parenthesis, so the original expression can be rewritten as:

We observe a telescoping pattern where the denominator of each fraction cancels with the numerator of the subsequent fraction:

After complete cancellation of adjacent terms, only the first numerator and the last denominator remain:

Final result:

 Observing the expression, we notice repeated numbers in each parenthesis.

Factoring out these common terms, we can rewrite the expression as:  

Rearranging the terms by moving the constants to the left and the fractions to the right, we get:  

The right part of the expression is a telescoping product:  

Therefore, the original expression simplifies to:  

Final result:

By observation, we notice that all denominators are perfect squares.

We can attempt the following transformation:

Now we group terms to make the cancellation explicit:

Each pair in parentheses cancels out completely:

After all cancellations, we are left with:

Final result:

By observing the expression, we notice that the denominators can be factored using the difference of squares formula .

Let's rewrite the expression accordingly:  

Now we can see a clear cancellation pattern emerging :

After complete cancellation of common factors, we are left with:  

Observing this telescoping product, we see that each numerator cancels with the denominator of the next fraction:  

After all cancellations, only the first numerator and the last denominator remain:  

Final result:

Let's evaluate the expression by first converting each term to an improper fraction:

Now we can write this product as a single fraction:

We observe a clear cancellation pattern:

After complete cancellation of common factors in numerator and denominator, we are left with:

Now we can simplify the remaining expression:

Final result:

We immediately notice that this problem combines two types of telescoping products.

Let's rewrite the original expression:

Now we convert each term to fraction form:

We observe two separate cancellation patterns - one in the increasing sequence and one in the decreasing sequence. For the increasing sequence: For the decreasing sequence:

Combining both cancellations, we are left with:

Multiplying the remaining terms gives us the final result:

Final result: