Hilton Johnson
08/20/2023 · Primary School
What is the sum of the given summation notation below? \( S=\sum_{x=1}^{3}(-7 x+9)^{3}-\left(9 x-x^{3}\right) \)
UpStudy ThothAI Solution
Tutor-Verified Answer
Step-by-step Solution
To evaluate the sum \( S = \sum_{x=1}^{3} \left( (-7x + 9)^3 - (9x - x^3) \right) \), we will first compute the expression inside the summation for each integer value of \( x \) from 1 to 3.
### Step 1: Calculate for \( x = 1 \)
\[
(-7(1) + 9)^3 - (9(1) - (1)^3) = (-7 + 9)^3 - (9 - 1)
\]
\[
= 2^3 - 8 = 8 - 8 = 0
\]
### Step 2: Calculate for \( x = 2 \)
\[
(-7(2) + 9)^3 - (9(2) - (2)^3) = (-14 + 9)^3 - (18 - 8)
\]
\[
= (-5)^3 - 10 = -125 - 10 = -135
\]
### Step 3: Calculate for \( x = 3 \)
\[
(-7(3) + 9)^3 - (9(3) - (3)^3) = (-21 + 9)^3 - (27 - 27)
\]
\[
= (-12)^3 - 0 = -1728 - 0 = -1728
\]
### Step 4: Sum the results
Now we sum the results from each calculation:
\[
S = 0 + (-135) + (-1728) = -135 - 1728 = -1863
\]
Thus, the final result is:
\[
\boxed{-1863}
\]
Quick Answer
The sum is -1863.
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