Q. 37 Find a formula for each of the s... [FREE SOLUTION]

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Chapter 4: Q. 37 (page 326)

Find a formula for each of the sums in Exercises 37, and then use these formulas to calculate each sum for $n = 100, 500$ and $1000$ .
${鈭�}_{k = 3}^{n} (k + 1)^{2}$

Short Answer

Expert verified

The formula is $\begin{array}{rcl} \frac{2 n^{3} + 9 n^{2} + 13 n}{6} \end{array}$ .

The sum when $n = 100$ is . $348550$

The sum when $n = 500$ is $\begin{array}{rcl} 42042750 \end{array}$ .

The sum when $n = 1000$ is $\begin{array}{rcl} 334835500 \end{array}$ .

Step by step solution

Given information

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

Determine the formula of the given summation

The sum can be written as:

$\begin{array}{rcl} {鈭�}_{k = 3}^{n} (k + 1)^{2} & = & {鈭�}_{k = 3}^{n} (k^{2} + 2 k + 1) \\ = & {鈭�}_{k = 3}^{n} k^{2} + 2 {鈭�}_{k = 3}^{n} k + {鈭�}_{k = 3}^{n} 1 \\ = & (\frac{n (n + 1) (2 n + 1)}{6}) + 2 \frac{n (n + 1)}{2} + n [{鈭�}_{k = 1}^{n} k^{2} = \frac{n (n + 1) (2 n + 1)}{6}, {鈭�}_{k = 1}^{n} k = \frac{n (n + 1)}{2}, {鈭�}_{k = 1}^{n} 1 = n] \\ = & \frac{2 n^{3} + 9 n^{2} + 13 n}{6} \end{array}$

Evaluate the sum for n=100,500 and 1000

Substitute $100$ for $n$ in $\begin{array}{rcl} \frac{2 n^{3} + 9 n^{2} + 13 n}{6} \end{array}$ .

$\begin{array}{rcl} \frac{2 {(100)}^{3} + 9 {(100)}^{2} + 13 (100)}{6} \end{array} = 348550$

Substitute $500$ for $n$ in $\begin{array}{rcl} \frac{2 n^{3} + 9 n^{2} + 13 n}{6} \end{array}$ .

$\begin{array}{rcl} \frac{2 {(500)}^{3} + 9 {(500)}^{2} + 13 (500)}{6} & = & 42042750 \end{array}$

Substitute $1000$ for $n$ in $\begin{array}{rcl} \frac{2 n^{3} + 9 n^{2} + 13 n}{6} \end{array}$ .

$\begin{array}{rcl} \frac{2 {(1000)}^{3} + 9 {(1000)}^{2} + 13 (1000)}{6} & = & 334835500 \end{array}$

Write the conclusion

The formula is $\begin{array}{rcl} \frac{2 n^{3} + 9 n^{2} + 13 n}{6} \end{array}$ .

The sum whenrole="math" localid="1649139413983" $n = 100, 500$ and $1000$ is $348550, 42042750$ and $334835500$ .

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91影视

Find a formula for each of the sums in Exercises 37, and then use these formulas to calculate each sum for $n = 100, 500$ and $1000$ .
${鈭�}_{k = 3}^{n} (k + 1)^{2}$

Short Answer

Step by step solution

Given information

Determine the formula of 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 37, and then use these formulas to calculate each sum for n=100,500and 1000.鈭�k=3n(k+1)2

Short Answer

Step by step solution

Given information

Determine the formula of 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

Calculus

Decision Maths

Pure Maths

Probability and Statistics

Applied Mathematics

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Find a formula for each of the sums in Exercises 37, and then use these formulas to calculate each sum for $n = 100, 500$ and $1000$ .
${鈭�}_{k = 3}^{n} (k + 1)^{2}$