Chapter 12: Q 75. (page 809)

In Problems 71-82, find the sum of each sequence.
${\sum (5}_{k = 1}^{20} k + 3)$

Short Answer

Expert verified

The sum of this sequence is 1110.

Step by step solution

Step 1. Write the given information.

The sum of sequence:

${\sum (5 k + 3)}_{k = 1}^{20}$

Step 2. Use the properties of sequence.

Using the properties of sequence:
${\sum (a_{k} + b_{k})}_{k = 1}^{n} = {\sum (a_{k})}_{k = 1}^{n} + {\sum (b_{k})}_{k = 1}^{n} & {\sum (c a_{k})}_{k = 1}^{n} = c {\sum a_{k}}_{k = 1}^{n}$

So,

${\sum (5 k + 3)}_{k = 1}^{20} = {\sum (5 k)}_{k = 1}^{20} + {\sum (3)}_{k = 1}^{20}$

${\sum (5 k)}_{k = 1}^{20} = {5 \sum (k)}_{k = 1}^{20}$

Step 3. Use the formula for sum of sequences of n real numbers.

Using formula for summation is:

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

So,

$5 {\sum k}_{k = 1}^{20} = 5 \times \frac{20 (20 + 1)}{2} \Rightarrow {5 \sum k}_{k = 1}^{n} = 5 \times \frac{20 (21)}{2} \Rightarrow {5 \sum k}_{k = 1}^{n} = 5 \times 210 \Rightarrow {5 \sum k}_{k = 1}^{n} = 1050$

Step 4. Use the formula for sum of sequence.

Using formula for summation is:

${\sum c}_{k = 1}^{n} = c + c + . . . + c \Rightarrow {\sum c}_{k = 1}^{n} = c n, c - r e a l n u m b e r S o, {\sum 3}_{k = 1}^{20} = 3 \times 20 \Rightarrow {\sum 3}_{k = 1}^{20} = 60$

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

From Step 3,

${5 \sum k}_{k = 1}^{n} = 1050$

From Step 4,

${\sum 3}_{k = 1}^{20} = 60$

So,

localid="1646750229919" ${\sum (5 k + 3)}_{k = 1}^{20} = {\sum (5 k)}_{k = 1}^{20} + {\sum (3)}_{k = 1}^{20} \Rightarrow {\sum (5 k + 3)}_{k = 1}^{20} = 1050 + 60 \Rightarrow {\sum (5 k + 3)}_{k = 1}^{20} = 1110$

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In Problems 71-82, find the sum of each sequence.
${\sum (5}_{k = 1}^{20} k + 3)$

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 n real numbers.

Step 4. Use the formula for sum of sequence.

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

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In Problems 71-82, find the sum of each sequence.∑(5k=120k+3)

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 n real numbers.

Step 4. Use the formula for sum of sequence.

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

Probability and Statistics

Calculus

Pure Maths

Discrete Mathematics

Theoretical and Mathematical Physics

Logic and Functions

Study anywhere. Anytime. Across all devices.

In Problems 71-82, find the sum of each sequence.
${\sum (5}_{k = 1}^{20} k + 3)$