Q. 36 Find a formula for each of the s... [FREE SOLUTION]

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Chapter 4: Q. 36 (page 326)

Find a formula for each of the sums in Exercises 36, and then use these formulas to calculate each sum for $n = 100, 500$ and $n = 1000$
${鈭�}_{k = 1}^{n} (k^{3} - 10 k^{2} + 2)$

Short Answer

Expert verified

The formula of the given summation is $\frac{3 n^{4} - 34 n^{3} - 57 n^{2} + 4 n}{12}$ .

The sum when $n = 100$ is $24716191700$ .

The sum when $n = 500$ is $15269646000$ .

The sum when $n = 1000$ is $247161917000$ .

Step by step solution

Given information

The given summation is ${鈭�}_{k = 1}^{n} (k^{3} - 10 k^{2} + 2)$ .

Determine the formula for the given summation

The sum can be written as:

${鈭�}_{k = 1}^{n} (k^{3} - 10 k^{2} + 2) = {鈭�}_{k = 1}^{n} k^{3} - 10 {鈭�}_{k = 1}^{n} k^{2} + 2 {鈭�}_{k = 1}^{n} 1 = \frac{n^{2} (n + 1)^{2}}{4} - 10 (\frac{n (n + 1) (2 n + 1)}{6}) + 2 n [{鈭�}_{k = 1}^{n} k^{3} = \frac{n^{2} (n + 1)^{2}}{4}, {鈭�}_{k = 1}^{n} k^{2} = \frac{n (n + 1) (2 n + 1)}{6}, {鈭�}_{k = 1}^{n} 1 = n] = \frac{3 n^{4} - 34 n^{3} - 57 n^{2} + 4 n}{12}$

Evaluate the sum for n=100,500 and 1000

Substitute $100$ for $n$ in $\frac{3 n^{4} - 34 n^{3} - 57 n^{2} + 4 n}{12}$ .

role="math" localid="1649073228160" $\frac{3 {(100)}^{4} - 34 {(100)}^{3} - 57 {(100)}^{2} + 4 (100)}{12} = 24716191700$

Substitute $500$ for $n$ in $\frac{3 n^{4} - 34 n^{3} - 57 n^{2} + 4 n}{12}$ .

role="math" localid="1649073142785" $\frac{3 {(500)}^{4} - 34 {(500)}^{3} - 57 {(500)}^{2} + 4 (500)}{12} = 15269646000$

Substitute $1000$ for $n$ in $\frac{3 n^{4} - 34 n^{3} - 57 n^{2} + 4 n}{12}$ .

role="math" localid="1649073220118" $\frac{3 {(1000)}^{4} - 34 {(1000)}^{3} - 57 {(1000)}^{2} + 4 (1000)}{12} = 247161917000$

Write the conclusion

The formula is $\frac{3 n^{4} - 34 n^{3} - 57 n^{2} + 4 n}{12}$ .

The sum when $n = 100, 500$ and $1000$ is $24716191700, 15269646000, and 247161917000$

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91影视

Find a formula for each of the sums in Exercises 36, and then use these formulas to calculate each sum for $n = 100, 500$ and $n = 1000$
${鈭�}_{k = 1}^{n} (k^{3} - 10 k^{2} + 2)$

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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91影视

Find a formula for each of the sums in Exercises 36, and then use these formulas to calculate each sum for n=100,500and n=1000鈭�k=1nk3-10k2+2

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

Discrete Mathematics

Geometry

Probability and Statistics

Logic and Functions

Mechanics Maths

Statistics

Study anywhere. Anytime. Across all devices.

Find a formula for each of the sums in Exercises 36, and then use these formulas to calculate each sum for $n = 100, 500$ and $n = 1000$
${鈭�}_{k = 1}^{n} (k^{3} - 10 k^{2} + 2)$