Math Problem Statement
Solution
Let's break down the given expression step by step.
The sum runs from to for the following expression:
Step-by-step breakdown:
-
First term inside the parentheses:
- Simplifying .
- The square root of this expression is .
- , because raising to the fourth root cancels the exponent of 4.
So, the first term simplifies to:
-
Second term inside the parentheses:
- Simplify , and .
- , so .
Therefore, the second term becomes:
-
Third term inside the parentheses:
- Since , we have .
This is a constant term.
Now, putting it all together:
The cancels out with , so the expression simplifies to:
Final summation:
Now, we need to sum this from to :
This can be split into two sums:
-
The sum of the first 20 integers is given by the formula , so for :
-
The sum of 20 constants (5) is just .
Now calculate the total sum:
Final Answer:
The value of the sum is 1780.
Would you like more details or have any questions about the steps?
Here are five related questions you could explore:
- How does the sum of an arithmetic sequence work?
- Can this method be extended to sums with different powers of ?
- How do logarithmic functions interact with sums?
- What is the difference between square roots and fourth roots?
- How would the result change if the upper limit of the sum was different?
Tip: Always simplify the terms inside the sum before performing the summation to avoid complicated expressions.
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Math Problem Analysis
Mathematical Concepts
Summation
Logarithms
Exponents
Roots
Formulas
Sum of integers: sum(i) from 1 to n = n(n+1)/2
Properties of logarithms: log_b(b^n) = n
Root simplifications: sqrt(a^n) = a^(n/2)
Theorems
Arithmetic sum formula
Logarithmic identities
Suitable Grade Level
College level or advanced high school
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