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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 12: Q 77. (page 809)

In Problems 71-82, find the sum of each sequence.
${鈭� (k^{2}}_{k = 1}^{16} + 4)$

Short Answer

Expert verified

The sum of this sequence is 1560.

Step by step solution

01

Step 1. Write the given information.

The sum of sequences:

${鈭� (k^{2} + 4)}_{k = 1}^{16} = {鈭� (k^{2})}_{k = 1}^{16} + {鈭� (4)}_{k = 1}^{16}$

02

Step 2. Use the properties of sequence.

Using the properties of sequence:

${鈭� (a_{k} + b_{k})}_{k = 1}^{n} = {鈭� (a_{k})}_{k = 1}^{n} + {鈭� (b_{k})}_{k = 1}^{n}$

So,

${鈭� (k^{2} + 4)}_{k = 1}^{16} = {鈭� (k^{2})}_{k = 1}^{16} + {鈭� (4)}_{k = 1}^{16}$

03

Step 3. Use the formula for sum of sequences of (n2 ) real numbers.

Using formula for summation is:

{鈭� k^{2}}_{k = 1}^{n} = 1^{2} + 2^{2} + . . . + n^{2} 鈬� {鈭� k^{2}}_{k = 1}^{n} = \frac{n (n + 1) (2 n + 1)}{6}

So,

${鈭� k^{2}}_{k = 1}^{16} = \frac{16 (16 + 1) ((2 脳 16) + 1)}{6} 鈬� {鈭� k^{2}}_{k = 1}^{16} = \frac{16 脳 17 脳 33}{6} 鈬� {鈭� k^{2}}_{k = 1}^{16} = \frac{8976}{6} 鈬� {鈭� k^{2}}_{k = 1}^{16} = 1496$

04

Step 4. Use the formula for sum of sequences.

Using formula for summation is:
${鈭� c}_{k = 1}^{n} = c + c + . . . + c 鈬� {鈭� c}_{k = 1}^{n} = c n, c - r e a l n u m b e r S o, {鈭� 4}_{k = 1}^{16} = 4 脳 16 鈬� {鈭� 4}_{k = 1}^{16} = 64$

05

Step 5. Now, add the two sums from Step 3 and Step 4.

From Step 3,

${鈭� k^{2}}_{k = 1}^{16} = 1496$

From Step 4,

${鈭� 4}_{k = 1}^{16} = 64$

So,

${鈭� (k^{2} + 4)}_{k = 1}^{16} = {鈭� (k^{2})}_{k = 1}^{16} + {鈭� (4)}_{k = 1}^{16} 鈬� {鈭� (k^{2} + 4)}_{k = 1}^{16} = 1496 + 64 鈬� {鈭� (k^{2} + 4)}_{k = 1}^{16} = 1560$

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Most popular questions from this chapter

In Problems 37鈥�50, a sequence is defined recursively. Write down the first five terms.

$a_{1} = - 2; a_{n} = n + a_{n - 1}$

In Problems 37鈥�50, a sequence is defined recursively. Write down the first five terms.

$a_{1} = A; a_{n} = r a_{n - 1}, r 鈮� 0$

Education IRA: On January 1, 1999, John鈥檚 parents decided to place \(45 at the end of each month into an Education IRA.

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(c) What will be the value of the account in 16 years when John goes to college?

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$\{S_{n}\} = \{\frac{2^{n}}{3^{n} + 1}\}$

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${鈭� k^{3}}_{k = 4}^{24}$

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