Chapter 12: Q 78. (page 809)

In Problems 71-82, find the sum of each sequence.
${\sum (k^{2}}_{k = 0}^{14} - 4)$

Short Answer

Expert verified

The sum of this sequence is 955.

Step by step solution

Step 1. Write the given information.

The sum of sequence:

${\sum (k^{2} - 4)}_{k = 0}^{14}$

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,

${\sum (k^{2} - 4)}_{k = 1}^{14} = {\sum (k^{2})}_{k = 1}^{14} {- \sum (4)}_{k = 1}^{14}$

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

Using formula for summation is:

${\sum k^{2}}_{k = 0}^{n} = 1^{2} + 2^{2} + . . . + n^{2} {\sum k^{2}}_{k = 0}^{n} = 0 + {\sum k^{2}}_{k = 1}^{n} \Rightarrow {\sum k^{2}}_{k = 0}^{n} = 0 + \frac{n (n + 1) (2 n + 1)}{6}$

So,

${\sum k^{2}}_{k = 0}^{14} = \frac{14 (14 + 1) ((2 \times 14) + 1)}{6} \Rightarrow {\sum k^{2}}_{k = 0}^{14} = \frac{6090}{6} \Rightarrow {\sum k^{2}}_{k = 0}^{14} = 1015$

Step 4. Use the formula for sum of sequences.

Using formula for summation is:

${\sum c}_{k = 0}^{n} = c + c + . . . + c \Rightarrow {\sum c}_{k = 0}^{n} = c n, c - r e a l n u m b e r S o, {\sum 4}_{k = 0}^{14} = 4 \times (14 + 1) \Rightarrow {\sum 4}_{k = 0}^{14} = 4 \times 15 \Rightarrow {\sum 4}_{k = 0}^{14} = 60$

Step 5. Now, subtract Step 3 from Step 4.

From Step 3,

${\sum k^{2}}_{k = 1}^{14} = 1015$

From Step 4,

${\sum 4}_{k = 1}^{14} = 60$

So,

${\sum (k^{2} - 4)}_{k = 1}^{14} = {\sum (k^{2})}_{k = 1}^{14} {- \sum (4)}_{k = 1}^{14} \Rightarrow {\sum (k^{2} - 4)}_{k = 1}^{14} = 1015 - 60 \Rightarrow {\sum (k^{2} - 4)}_{k = 1}^{14} = 955$

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In Problems 71-82, find the sum of each sequence.
${\sum (k^{2}}_{k = 0}^{14} - 4)$

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 ), n - real numbers.

Step 4. Use the formula for sum of sequences.

Step 5. Now, subtract Step 3 from Step 4.

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In Problems 71-82, find the sum of each sequence.∑(k2k=014-4)

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 ), n - real numbers.

Step 4. Use the formula for sum of sequences.

Step 5. Now, subtract Step 3 from Step 4.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Statistics

Mechanics Maths

Geometry

Pure Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

In Problems 71-82, find the sum of each sequence.
${\sum (k^{2}}_{k = 0}^{14} - 4)$