Q. 28 Evaluate the sums in Exercises 2... [FREE SOLUTION]

91影视

Chapter 13: Q. 28 (page 1004)

Evaluate the sums in Exercises $23 鈥� 28$
${鈭�}_{i = 1}^{l} {鈭�}_{j = 1}^{m} {鈭�}_{k = 1}^{m} i j^{2} k^{3}$

Short Answer

Expert verified

The value of summation is

$\frac{l m n^{2} (l + 1) (m + 1) (2 m + 1) (n + 1)^{2}}{48}$

Step by step solution

Step 1. Given information

An expression s given as ${鈭�}_{i = 1}^{l} {鈭�}_{j = 1}^{m} {鈭�}_{k = 1}^{m} i j^{2} k^{3}$

Step 2. Evaluating sum

Separate the first to last as,

${鈭�}_{i = 1}^{l} {鈭�}_{j = 1}^{m} {鈭�}_{k = 1}^{k} i j^{2} k^{3} = {鈭�}_{i = 1}^{l} {鈭�}_{j = 1}^{m} ({鈭�}_{i = 1}^{n} i j^{2} k^{3})$

Now simplify the summation as,

${鈭�}_{i = 1}^{n} i j^{2} k^{3} = i j^{2} 1^{3} + i j^{2} 2^{3} + i j^{2} 3^{3} + 鈥� + i j^{2} n^{3} = i j^{2} (1^{3} + 2^{3} + 3^{3} + 鈥� + n^{3}) = i j^{2} \frac{n^{2} (n + 1)^{2}}{4}$

Now simplify the each terms as above,

${鈭�}_{k = 1}^{1} {鈭�}_{j = 1}^{m} {鈭�}_{k = 1}^{n} i j^{2} k^{3} = {鈭�}_{i = 1}^{1} {鈭�}_{j = 1}^{m} ({鈭�}_{k = 1}^{n} i j^{2} k^{3}) = {鈭�}_{j = 1}^{1} ({鈭�}_{j = 1}^{m} i j^{2} \frac{n^{2} (n + 1)^{2}}{4}) {鈭�}_{j = 1}^{m} i^{2} \frac{n^{2} (n + 1)^{2}}{4} = i \frac{n^{2} (n + 1)^{2}}{4} ({鈭�}_{j = 1}^{m} j^{2}) = i \frac{n^{2} (n + 1)^{2}}{4} (1^{2} + 2^{2} + 3^{2} + 鈥� + m^{2}) = i 脳 \frac{n^{2} (n + 1)^{2}}{4} 脳 \frac{m (m + 1) (2 m + 1)}{6} {鈭�}_{i = 1}^{1} {鈭�}_{j = 1}^{m} {鈭�}_{k = 1}^{n} i j^{2} k^{3} = {鈭�}_{i = 1}^{1} ({鈭�}_{j = 1}^{m} i j^{2} \frac{n^{2} (n + 1)^{2}}{4}) = {鈭�}_{i = 1}^{l} (i 脳 \frac{n^{2} (n + 1)^{2}}{4} 脳 \frac{m (m + 1) (2 m + 1)}{6}) = \frac{n^{2} (n + 1)^{2}}{4} 脳 \frac{m (m + 1) (2 m + 1)}{6} 脳 ({鈭�}_{i = 1}^{1} (i)) = \frac{n^{2} (n + 1)^{2}}{4} 脳 \frac{m (m + 1) (2 m + 1)}{6} 脳 (1 + 2 + 3 + 鈥� + l) = \frac{n^{2} (n + 1)^{2}}{4} 脳 \frac{m (m + 1) (2 m + 1)}{6} 脳 \frac{l (l + 1)}{2} = \frac{l m n^{2} (l + 1) (m + 1) (2 m + 1) (n + 1)^{2}}{48}$

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.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Evaluate the sums in Exercises $23 鈥� 28$
${鈭�}_{i = 1}^{l} {鈭�}_{j = 1}^{m} {鈭�}_{k = 1}^{m} i j^{2} k^{3}$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Evaluating sum

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Applied Mathematics

Decision Maths

Pure Maths

Geometry

Logic and Functions

Study anywhere. Anytime. Across all devices.

91影视

Evaluate the sums in Exercises 23鈥�28鈭�i=1l鈭�j=1m鈭�k=1mij2k3

Short Answer

Step by step solution

Step 1. Given information

Step 2. Evaluating sum

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Applied Mathematics

Decision Maths

Pure Maths

Geometry

Logic and Functions

Study anywhere. Anytime. Across all devices.

Evaluate the sums in Exercises $23 鈥� 28$
${鈭�}_{i = 1}^{l} {鈭�}_{j = 1}^{m} {鈭�}_{k = 1}^{m} i j^{2} k^{3}$