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Chapter 4: Q. 38 (page 326)

Find a formula for each of the sums in Exercises 38, and then use these formulas to calculate each sum for $n = 100, 500$ and $1000$

Short Answer

Expert verified

The formula of given summation is $\frac{- 4 n + n^{4} + 2 n^{3} + n^{2}}{16}$ .

The sum when $n = 100$ is $6375600$ .

The sum when $n = 500$ is $3921890500$ .

The sum when $n = 1000$ is $62625062250$ .

Step by step solution

01

Given information

The given summation is ${鈭�}_{k = 1}^{n} \frac{k^{3} - 1}{4}$ .

02

Determine the formula for the given summation.

The sum can be written as:

${鈭�}_{k = 1}^{n} \frac{k^{3} - 1}{4} = \frac{1}{4} {鈭�}_{k = 1}^{n} (k^{3} - 1) = \frac{1}{4} ({鈭�}_{k = 1}^{n} k^{3} - {鈭�}_{k = 1}^{n} 1) = \frac{1}{4} (\frac{n^{2} (n + 1)^{2}}{4} - n) = \frac{- 4 n + n^{4} + 2 n^{3} + n^{2}}{16}$

03

Evaluate the sum for n=100,500 and 1000

Substitute $100$ for $n$ in $\frac{- 4 n + n^{4} + 2 n^{3} + n^{2}}{16}$ .

$\frac{- 4 (100) + {(100)}^{4} + 2 {(100)}^{3} + {(100)}^{2}}{16} = 6375600$

Substitute $500$ for $n$ in $\frac{- 4 n + n^{4} + 2 n^{3} + n^{2}}{16}$ .

$\frac{- 4 (500) + {(500)}^{4} + 2 {(500)}^{3} + {(500)}^{2}}{16} = 3921890500$

Substitute $1000$ for $n$ in $\frac{- 4 n + n^{4} + 2 n^{3} + n^{2}}{16}$ .

$\frac{- 4 (1000) + {(1000)}^{4} + 2 {(1000)}^{3} + {(1000)}^{2}}{16} = 62625062250$

04

Write the conclusion

The formula is $\frac{- 4 n + n^{4} + 2 n^{3} + n^{2}}{16}$ .

The sum when $n = 100, 500,$ and $1000$ arelocalid="1649222788112" $6375600, 3921890500,$ and $62625062250$ .

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Most popular questions from this chapter

Find the sum or quantity without completely expanding or calculating any sums.

Given ${鈭�}_{k = 1}^{4} a_{k} = 7$ and $, {鈭�}_{k = 0}^{4} b_{k} = 10$ , $a_{0} = 2$ . Find the value of ${鈭�}_{k = 0}^{4} (2 a_{k} + 3 b_{k})$ .

Show that $F (x) = \sin x - x \cos x + 2$ is an antiderivative of $f (x) = \sin x$ .

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${鈭�}_{- 2}^{- 2} x (f (x) + 3)^{2} d x$

Prove Theorem 4.13(c): For any real numbers a and b, ${鈭�}_{a}^{b} x^{2} d x = \frac{1}{3} (b^{3} - a^{3}) .$ Use the proof of Theorem 4.13(a) as a guide.

Suppose f is positive on (鈭掆垶, 鈭�1] and [2,鈭�) and negative on the interval [鈭�1, 2]. Write (a) the signed area and (b) the absolute area between the graph of f and the x-axis on [鈭�3, 4] in terms of definite integrals that do not involve absolute values.

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