[tex]\sum \frac{7}{11}-\frac{7}{8}[/tex]
Answer (1)
Simplify the expression inside the summation: 11 7 − 8 7 .
Find a common denominator (88) and rewrite the fractions: 88 56 − 88 77 .
Subtract the fractions: 88 56 − 77 = 88 − 21 .
If the summation is over n terms, the result is 88 − 21 n . Assuming n = 1 , the final answer is − 88 21 .
Explanation
Understanding the Problem We are asked to evaluate the expression ∑ 11 7 − 8 7 . The summation symbol is not properly defined. We assume that we are summing a constant value. We also assume that the summation is over an index that is not specified. Without the limits of summation, we cannot evaluate the expression exactly. However, we can simplify the expression inside the summation first.
Finding a Common Denominator First, let's simplify the expression 11 7 − 8 7 . To do this, we need to find a common denominator, which is the least common multiple (LCM) of 11 and 8. Since 11 and 8 have no common factors, their LCM is simply their product, which is 11 × 8 = 88 . Now we can rewrite the fractions with the common denominator:
Subtracting the Fractions 11 7 = 11 × 8 7 × 8 = 88 56 and 8 7 = 8 × 11 7 × 11 = 88 77 . Therefore, 11 7 − 8 7 = 88 56 − 88 77 = 88 56 − 77 = 88 − 21 .
Summing the Constant Value Now, let's assume the summation is over n terms. Then, the expression becomes ∑ i = 1 n ( 11 7 − 8 7 ) = ∑ i = 1 n 88 − 21 = n × 88 − 21 = 88 − 21 n . Since we don't know the value of n , we can only simplify the expression to this point. If n = 1 , then the result is 88 − 21 .
Final Answer Without knowing the limits of the summation, we can only simplify the expression to 88 − 21 n , where n is the number of terms in the summation. If we assume n = 1 , the result is − 88 21 .
Examples
Imagine you are calculating the net change in your investment portfolio. If, on average, each investment changes by 11 7 − 8 7 units daily, summing these changes over a period helps determine the overall gain or loss. Understanding how to simplify and sum such expressions is crucial in finance for tracking performance and making informed decisions. This skill is also applicable in various fields like physics, where summing small changes over time can describe motion or energy transfer.
