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Chapter 14: Problem 78

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\sum_{i=1}^{2} a_{i} b_{i}=\sum_{i=1}^{2} a_{i} \sum_{i=1}^{2} b_{i}$$

Short Answer

Expert verified
The statement is false. The correct statement should be \[\sum_{i=1}^{2} a_{i} b_{i} = a_1 \cdot b_1 + a_2 \cdot b_2\] when we're iterating through terms of series, we sum the products individually, it cannot be simplified to \[\left(a_1 + a_2\right) \cdot \left(b_1 + b_2\right)\].

Step by step solution

01

Evaluate the left-hand side of the statement

The left-hand side denotes the product of each term of two sequences, from i=1 to i=2. This would mean computing \[a_1 \cdot b_1 + a_2 \cdot b_2\]
02

Evaluate the right-hand side of the statement

The right-hand side denotes the sum of the sequences from i=1 to 2, each multiplied by the other. This would compute to \[\left(a_1 + a_2\right) \cdot \left(b_1 + b_2\right)\]
03

Compare both sides

Since \[a_1 \cdot b_1 + a_2 \cdot b_2\] is not always equal to \[\left(a_1 + a_2\right) \cdot \left(b_1 + b_2\right)\], the statement cannot always hold true. The false assumption lies in the belief that one can compute the sums of the sequences first and then the product, similar to the distributive property.
04

Correct the statement to make it true

To make the statement always true, it should be adjusted as follows \[\sum_{i=1}^{2} a_{i} b_{i} = a_1 \cdot b_1 + a_2 \cdot b_2\] and not equal to \[\left(a_1 + a_2\right) \cdot \left(b_1 + b_2\right)\]

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Most popular questions from this chapter

Find the term in the expansion of \(\left(x^{2}+y^{2}\right)^{5}\) containing \(x^{4}\) as a factor.

If \(f(x)=x^{2}+2 x+3,\) find \(f(a+1)\) (Section \(8.1,\) Example 3 )

Each exercise involves observing a pattern in the expanded form of the binomial expression \((a+b)^{n}\).$$\begin{array}{l}(a+b)^{1}=a+b \\\\(a+b)^{2}=a^{2}+2 a b+b^{2} \\\\(a+b)^{3}=a^{3}+3 a^{2} b+3 a b^{2}+b^{3} \\\\(a+b)^{4}=a^{4}+4 a^{3} b+6 a^{2} b^{2}+4 a b^{3}+b^{4} \\\\(a+b)^{5}=a^{5}+5 a^{4} b+10 a^{3} b^{2}+10 a^{2} b^{3}+5 a b^{4}+b^{5}\end{array}$$ Describe the pattern for the sum of the exponents on the variables in each term.

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