Q. 26 Evaluate the sums in Exercises 2... [FREE SOLUTION]

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Chapter 13: Q. 26 (page 1004)

Evaluate the sums in Exercises $23 鈥� 28$ .
${鈭�}_{j = 1}^{m} {鈭�}_{k = 1}^{n} (j - k)$

Short Answer

Expert verified

The value of the summation is $\frac{1}{2} m n (m - n)$

Step by step solution

Step 1. Given information

An expression is given as ${鈭�}_{j = 1}^{m} {鈭�}_{k = 1}^{n} (j - k)$

Step 2. Evaluating summation

First separate out the term then simplify,

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

Now simplify the other terms in the same manner,

${鈭�}_{j = 1}^{m} {鈭�}_{k = 1}^{n} (j - k) = {鈭�}_{j = 1}^{m} ({鈭�}_{k = 1}^{n} (j - k)) = {鈭�}_{j = 1}^{m} (n 脳 j - \frac{n (n + 1)}{2}) = (n 脳 1 - \frac{n (n + 1)}{2}) + (n 脳 2 - \frac{n (n + 1)}{2}) + (n 脳 3 - \frac{n (n + 1)}{2}) + . . . . . . . + (n 脳 m - \frac{n (n + 1)}{2}) = n (1 + 2 + 3 + 鈥� + m) - m 脳 \frac{n (n + 1)}{2} = n 脳 \frac{m (m + 1)}{2} - m 脳 \frac{n (n + 1)}{2} = m n (\frac{m + 1}{2} - \frac{n + 1}{2}) = m n (\frac{m - n}{2}) = \frac{1}{2} m n (m - n)$

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91影视

Evaluate the sums in Exercises $23 鈥� 28$ .
${鈭�}_{j = 1}^{m} {鈭�}_{k = 1}^{n} (j - k)$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Evaluating summation

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91影视

Evaluate the sums in Exercises 23鈥�28.鈭�j=1m鈭�k=1n(j-k)

Short Answer

Step by step solution

Step 1. Given information

Step 2. Evaluating summation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Calculus

Statistics

Theoretical and Mathematical Physics

Geometry

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Evaluate the sums in Exercises $23 鈥� 28$ .
${鈭�}_{j = 1}^{m} {鈭�}_{k = 1}^{n} (j - k)$