Chapter 12: Q 77. (page 809)

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

Short Answer

Expert verified

The sum of this sequence is 1560.

Step by step solution

Step 1. Write the given information.

The sum of sequences:

${\sum (k^{2} + 4)}_{k = 1}^{16} = {\sum (k^{2})}_{k = 1}^{16} + {\sum (4)}_{k = 1}^{16}$

Step 2. Use the properties of sequence.

Using the properties of sequence:

${\sum (a_{k} + b_{k})}_{k = 1}^{n} = {\sum (a_{k})}_{k = 1}^{n} + {\sum (b_{k})}_{k = 1}^{n}$

So,

${\sum (k^{2} + 4)}_{k = 1}^{16} = {\sum (k^{2})}_{k = 1}^{16} + {\sum (4)}_{k = 1}^{16}$

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

Using formula for summation is:

{\sum k^{2}}_{k = 1}^{n} = 1^{2} + 2^{2} + . . . + n^{2} \Rightarrow {\sum k^{2}}_{k = 1}^{n} = \frac{n (n + 1) (2 n + 1)}{6}

So,

${\sum k^{2}}_{k = 1}^{16} = \frac{16 (16 + 1) ((2 \times 16) + 1)}{6} \Rightarrow {\sum k^{2}}_{k = 1}^{16} = \frac{16 \times 17 \times 33}{6} \Rightarrow {\sum k^{2}}_{k = 1}^{16} = \frac{8976}{6} \Rightarrow {\sum k^{2}}_{k = 1}^{16} = 1496$

Step 4. Use the formula for sum of sequences.

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

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

From Step 3,

${\sum k^{2}}_{k = 1}^{16} = 1496$

From Step 4,

${\sum 4}_{k = 1}^{16} = 64$

So,

${\sum (k^{2} + 4)}_{k = 1}^{16} = {\sum (k^{2})}_{k = 1}^{16} + {\sum (4)}_{k = 1}^{16} \Rightarrow {\sum (k^{2} + 4)}_{k = 1}^{16} = 1496 + 64 \Rightarrow {\sum (k^{2} + 4)}_{k = 1}^{16} = 1560$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

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

Short Answer

Step by step solution

Step 1. Write the given information.

Step 2. Use the properties of sequence.

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

Step 4. Use the formula for sum of sequences.

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

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Calculus

Pure Maths

Probability and Statistics

Theoretical and Mathematical Physics

Decision Maths

Study anywhere. Anytime. Across all devices.

In Problems 71-82, find the sum of each sequence.∑(k2k=116+4)

Short Answer

Step by step solution

Step 1. Write the given information.

Step 2. Use the properties of sequence.

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

Step 4. Use the formula for sum of sequences.

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

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Calculus

Pure Maths

Probability and Statistics

Theoretical and Mathematical Physics

Decision Maths

Study anywhere. Anytime. Across all devices.

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