Math Problem Statement

Find the sum from i = 1 to 20 of the expression involving x^2, roots, logarithms, and powers.

Solution

Let's break down the given expression step by step.

The sum runs from i=1i = 1 to i=20i = 20 for the following expression:

x2(9x4x8i44+1255ilog55+log3243)x^2 \left( \sqrt{\frac{9x^4}{x^8}} \sqrt[4]{i^4} + \sqrt{\frac{125}{5}} i^{\log_5 5} + \log_3 243 \right)

Step-by-step breakdown:

  1. First term inside the parentheses: 9x4x8i44\sqrt{\frac{9x^4}{x^8}} \sqrt[4]{i^4}

    • Simplifying 9x4x8=9x4\frac{9x^4}{x^8} = \frac{9}{x^4}.
    • The square root of this expression is 9x4=3x2\sqrt{\frac{9}{x^4}} = \frac{3}{x^2}.
    • i44=i\sqrt[4]{i^4} = i, because raising to the fourth root cancels the exponent of 4.

    So, the first term simplifies to: 3ix2\frac{3i}{x^2}

  2. Second term inside the parentheses: 1255ilog55\sqrt{\frac{125}{5}} i^{\log_5 5}

    • Simplify 1255=25\frac{125}{5} = 25, and 25=5\sqrt{25} = 5.
    • log55=1\log_5 5 = 1, so ilog55=i1=ii^{\log_5 5} = i^1 = i.

    Therefore, the second term becomes: 5i5i

  3. Third term inside the parentheses: log3243\log_3 243

    • Since 243=35243 = 3^5, we have log3243=5\log_3 243 = 5.

    This is a constant term.

Now, putting it all together:

x2(3ix2+5i+5)x^2 \left( \frac{3i}{x^2} + 5i + 5 \right)

The x2x^2 cancels out with 3ix2\frac{3i}{x^2}, so the expression simplifies to:

3i+5i+5=8i+53i + 5i + 5 = 8i + 5

Final summation:

Now, we need to sum this from i=1i = 1 to i=20i = 20:

i=120(8i+5)\sum_{i=1}^{20} (8i + 5)

This can be split into two sums:

8i=120i+i=12058 \sum_{i=1}^{20} i + \sum_{i=1}^{20} 5

  • The sum of the first 20 integers is given by the formula i=1ni=n(n+1)2\sum_{i=1}^{n} i = \frac{n(n+1)}{2}, so for n=20n = 20:

    i=120i=20(21)2=210\sum_{i=1}^{20} i = \frac{20(21)}{2} = 210

  • The sum of 20 constants (5) is just 5×20=1005 \times 20 = 100.

Now calculate the total sum:

8×210+100=1680+100=17808 \times 210 + 100 = 1680 + 100 = 1780

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:

  1. How does the sum of an arithmetic sequence work?
  2. Can this method be extended to sums with different powers of ii?
  3. How do logarithmic functions interact with sums?
  4. What is the difference between square roots and fourth roots?
  5. 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