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Chapter 8: Problem 32

Find each indicated sum. $$\sum_{i=1}^{5} i^{3}$$

Short Answer

Expert verified
The sum of the cubes of the integers from 1 to 5 is 225.

Step by step solution

01

Identify the summation notation

Firstly, identify the summation notation. Here, it is denoted by \(\sum_{i=1}^{5} i^{3}\). This tells us to sum up, for \(i\) ranging from 1 to 5, the cube of \(i\). The 'i' under the summation symbol indicates the variable that will be changing and the 1 and 5 above and below are the starting and ending points respectively.
02

Calculate the cube of each integer

Next, calculate the cube of each integer from 1 to 5, individually.\[1^{3} = 1 \]\[2^{3} = 8\]\[3^{3} = 27\]\[4^{3} = 64\]\[5^{3} = 125\]
03

Compute the sum

The last step is to sum up all the calculated values.\[1 + 8 + 27 + 64 + 125 = 225\]

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Most popular questions from this chapter

Find the term indicated in each expansion. \(\left(x-\frac{1}{2}\right)^{9} ;\) fourth term

Graph each of the functions in the same viewing rectangle. Describe how the graphs illustrate the Binomial Theorem. $$ \begin{array}{l}f_{1}(x)=(x+1)^{4} \\\f_{2}(x)=x^{4} \\\f_{3}(x)=x^{4}+4 x^{3} \\\f_{4}(x)=x^{4}+4 x^{3}+6 x^{2} \\\f_{5}(x)=x^{4}+4 x^{3}+6 x^{2}+4 x \\\f_{6}(x)=x^{4}+4 x^{3}+6 x^{2}+4 x+1\end{array} $$ Use a \([-5,5,1]\) by \([-30,30,10]\) viewing rectangle.

Write the first three terms in each binomial expansion, expressing the result in simplified form. $$ (x-2 y)^{9} $$

What is the common ratio in a geometric sequence?

Describe the pattern on the exponents on \(b\) in the expansion of \((a+b)^{n}\).
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