Chapter 4: Q. 40 (page 326)

Find a formula for each of the sums in Exercises 40, and then use these formulas to calculate each sum for $n = 100, 500$ and $1000$

Short Answer

Expert verified

The formula of the given summation is $\frac{n^{2} + 3 n + 5}{3 n^{2}}$ .

The sum when $n = 100$ is $0.3435$ .

The sum when $n = 500$ is $0.33534$ .

The sum when $n = 1000$ is $0.334335$ .

Step by step solution

Given information

The given summation is . $\sum_{k = 1}^{n} \frac{k^{2} + k + 1}{n^{3}}$

Determine the formula for the given summation.

The sum can be written as:

$\sum_{k = 1}^{n} \frac{k^{2} + k + 1}{n^{3}} = \frac{1}{n^{3}} \sum_{k = 1}^{n} (k^{2} + k + 1) = \frac{1}{n^{3}} (\sum_{k = 1}^{n} k^{2} + \sum_{k = 1}^{n} k + \sum_{k = 1}^{n} 1) = \frac{1}{n^{3}} [(\frac{n (n + 1) (2 n + 1)}{6}) + \frac{n (n + 1)}{2} + n] = \frac{n^{2} + 3 n + 5}{3 n^{2}}$

Evaluate the sum for n=100,500 and 1000

Substitute $100$ for $n$ in $\frac{n^{2} + 3 n + 5}{3 n^{2}}$ .

role="math" localid="1649225056563" $\frac{{(100)}^{2} + 3 (100) + 5}{3 {(100)}^{2}} = 0.3435$

Substitute $500$ for $n$ in $\frac{n^{2} + 3 n + 5}{3 n^{2}}$ .

role="math" localid="1649225092265" $\frac{{(500)}^{2} + 3 (500) + 5}{3 {(500)}^{2}} = 0.33534$

Substitute $1000$ for $n$ in $\frac{n^{2} + 3 n + 5}{3 n^{2}}$ .

role="math" localid="1649225111347" $\frac{{(1000)}^{2} + 3 (1000) + 5}{3 {(1000)}^{2}} = 0.334335$

Write the conclusion

The formula is $\frac{n^{2} + 3 n + 5}{3 n^{2}}$ .

The sum when $n = 100, 500$ and $1000$ are $0.3435, 0.33534$ and $0.334335$ .

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Find a formula for each of the sums in Exercises 40, and then use these formulas to calculate each sum for $n = 100, 500$ and $1000$

Short Answer

Step by step solution

Given information

Determine the formula for the given summation.

Evaluate the sum for n=100,500 and 1000

Write the conclusion

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Find a formula for each of the sums in Exercises 40, and then use these formulas to calculate each sum for n=100,500and1000

Short Answer

Step by step solution

Given information

Determine the formula for the given summation.

Evaluate the sum for n=100,500 and 1000

Write the conclusion

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Theoretical and Mathematical Physics

Geometry

Mechanics Maths

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Probability and Statistics

Study anywhere. Anytime. Across all devices.

Find a formula for each of the sums in Exercises 40, and then use these formulas to calculate each sum for $n = 100, 500$ and $1000$