Chapter 13: Q. 4 (page 1003)

$Express the sum \frac{2}{5} + \frac{2}{6} + \frac{2}{7} + \frac{2}{8} + \frac{4}{5} + \frac{4}{6} + \frac{4}{7} + \frac{4}{8} using double - summation notation .$

Short Answer

Expert verified

$\frac{2}{5} + \frac{2}{6} + \frac{2}{7} + \frac{2}{8} + \frac{4}{5} + \frac{4}{6} + \frac{4}{7} + \frac{4}{8} = {鈭�}_{j = 1}^{j = 2} {鈭�}_{k = 5}^{k = 8} \frac{2^{j}}{k}$

Step by step solution

Step 1. Given Information

$\frac{2}{5} + \frac{2}{6} + \frac{2}{7} + \frac{2}{8} + \frac{4}{5} + \frac{4}{6} + \frac{4}{7} + \frac{4}{8} in double - summation notation .$

Step 2. Finding common terms and converting to summation

$\frac{2}{5} + \frac{2}{6} + \frac{2}{7} + \frac{2}{8} + \frac{4}{5} + \frac{4}{6} + \frac{4}{7} + \frac{4}{8} = (\frac{2}{5} + \frac{2}{6} + \frac{2}{7} + \frac{2}{8}) + (\frac{4}{5} + \frac{4}{6} + \frac{4}{7} + \frac{4}{8}) = (\frac{2^{1}}{5} + \frac{2^{1}}{6} + \frac{2^{1}}{7} + \frac{2^{1}}{8}) + (\frac{2^{2}}{5} + \frac{2^{2}}{6} + \frac{2^{2}}{7} + \frac{2^{2}}{8}) The common term is \frac{2^{j}}{5} + \frac{2^{j}}{6} + \frac{2^{j}}{7} + \frac{2^{j}}{8} and ranges from 1 to 2, which can be converted as \frac{2}{5} + \frac{2}{6} + \frac{2}{7} + \frac{2}{8} + \frac{4}{5} + \frac{4}{6} + \frac{4}{7} + \frac{4}{8} = (\frac{2^{1}}{5} + \frac{2^{1}}{6} + \frac{2^{1}}{7} + \frac{2^{1}}{8}) + (\frac{2^{2}}{5} + \frac{2^{2}}{6} + \frac{2^{2}}{7} + \frac{2^{2}}{8}) = {鈭�}_{j = 1}^{2} \frac{2^{j}}{5} + \frac{2^{j}}{6} + \frac{2^{j}}{7} + \frac{2^{j}}{8} The common term is \frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8} and ranges from 5 to 8, which can be converted as \frac{2}{5} + \frac{2}{6} + \frac{2}{7} + \frac{2}{8} + \frac{4}{5} + \frac{4}{6} + \frac{4}{7} + \frac{4}{8} = {鈭�}_{j = 1}^{2} \frac{2^{j}}{5} + \frac{2^{j}}{6} + \frac{2^{j}}{7} + \frac{2^{j}}{8} = {鈭�}_{j = 1}^{j = 2} {鈭�}_{k = 5}^{k = 8} \frac{2^{j}}{k} Hence, \frac{2}{5} + \frac{2}{6} + \frac{2}{7} + \frac{2}{8} + \frac{4}{5} + \frac{4}{6} + \frac{4}{7} + \frac{4}{8} = {鈭�}_{j = 1}^{j = 2} {鈭�}_{k = 5}^{k = 8} \frac{2^{j}}{k}$

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$Express the sum \frac{2}{5} + \frac{2}{6} + \frac{2}{7} + \frac{2}{8} + \frac{4}{5} + \frac{4}{6} + \frac{4}{7} + \frac{4}{8} using double - summation notation .$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Finding common terms and converting to summation

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Expressthesum25+26+27+28+45+46+47+48usingdouble-summationnotation.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Finding common terms and converting to summation

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Most popular questions from this chapter

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Discrete Mathematics

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Geometry

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$Express the sum \frac{2}{5} + \frac{2}{6} + \frac{2}{7} + \frac{2}{8} + \frac{4}{5} + \frac{4}{6} + \frac{4}{7} + \frac{4}{8} using double - summation notation .$