Chapter 13: Q. 8 (page 1003)

How many summands are in $\sum_{i = i_{0}}^{l} \sum_{j = j_{0}}^{m} \sum_{k = k_{0}}^{n} k^{i} j^{k}$ ?

Short Answer

Expert verified

Ans: The total number of summands in $\sum_{i = i_{0}}^{l} \sum_{j = j_{0}}^{m} \sum_{k = k_{0}}^{n} k^{i} j^{k}$ is $(k_{0} - n + 1) \times (j_{0} - m + 1)$

Step by step solution

Step 1. Given information:

$\sum_{i = i_{0}}^{l} \sum_{j = j_{0}}^{m} \sum_{k = k_{0}}^{n} k^{i} j^{k}$

Step 2. First Summation:

$\sum_{i = i_{0}}^{i = l} \sum_{j = j_{0}}^{j = m} \sum_{k = k_{0}}^{k = n} k^{i} j^{k} = \sum_{i = i_{0}}^{i = l} (\sum_{j = j_{0}}^{j = m} \sum_{k = k_{0}}^{k = n} k^{i} j^{k}) = \sum_{i = i_{0}}^{i = l} (\sum_{j = j_{0}}^{j = m} (\sum_{k = k_{0}}^{k = n} k^{i} j^{k}))$

The summation $\sum_{k = k_{0}}^{k = n} k^{i} j^{k}$ can be expanded by replacing k with $k_{0}$ and rewriting each term when k ends at n.

Hence, the number of terms in the summation $\sum_{k = k_{0}}^{k = n} k^{i} j^{k}$ is $k_{0} - n + 1$

Step 3. Second Summation:

Similarly, when expanded $\sum_{j = j_{0}}^{j = m} (\sum_{k = k_{0}}^{k = n} k^{i} j^{k})$ by replacing j from $j_{0}$ to m, the number of $\sum_{k = k_{0}}^{k = n} k^{i} j^{k}$ terms obtained is $j_{0} - m + 1$

Hence, the total number of summands of $\sum_{j = j_{0}}^{j = m} (\sum_{k = k_{0}}^{k = n} k^{i} j^{k})$ is $(k_{0} - n + 1) \times (j_{0} - m + 1)$

Step 4. Third Summation:

Similarly, when expanded $\sum_{i = i_{0}}^{i = l} (\sum_{j = j_{0}}^{j = m} (\sum_{k = k_{0}}^{k = n} k^{i} j^{k}))$ by replacing i from $i_{0}$ to l, the number of $\sum_{j = j_{0}}^{j = m} (\sum_{k = k_{0}}^{k = n} k^{i} j^{k})$ terms obtained is $i_{0} - l + 1$

The total number of summands in $\sum_{i = b_{0}}^{i = l} \sum_{j = j_{0}}^{j = m} \sum_{k = k_{0}}^{k = n} k^{i} j^{k}$ is $(i_{0} - l + 1) \times (k_{0} - n + 1) \times (j_{0} - m + 1)$

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.

How many summands are in $\sum_{i = i_{0}}^{l} \sum_{j = j_{0}}^{m} \sum_{k = k_{0}}^{n} k^{i} j^{k}$ ?

Short Answer

Step by step solution

Step 1. Given information:

Step 2. First Summation:

Step 3. Second Summation:

Step 4. Third Summation:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Theoretical and Mathematical Physics

Applied Mathematics

Mechanics Maths

Discrete Mathematics

Statistics

Study anywhere. Anytime. Across all devices.

How many summands are in ∑i=i0l∑j=j0m∑k=k0nkijk?

Short Answer

Step by step solution

Step 1. Given information:

Step 2. First Summation:

Step 3. Second Summation:

Step 4. Third Summation:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Theoretical and Mathematical Physics

Applied Mathematics

Mechanics Maths

Discrete Mathematics

Statistics

Study anywhere. Anytime. Across all devices.

How many summands are in $\sum_{i = i_{0}}^{l} \sum_{j = j_{0}}^{m} \sum_{k = k_{0}}^{n} k^{i} j^{k}$ ?