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Chapter 9: Problem 126

Determine whether the statement is true or false. Justify your answer. $$\sum_{j=1}^{4} 2^{j}=\sum_{j=3}^{6} 2^{j-2}$$

Short Answer

Expert verified
The statement is true as after computing, both sides yield the same result, 30.

Step by step solution

01

Calculate the left side sum

We calculate the sum on the left side of the equation. We generate terms by substituting \(j\) with values from 1 to 4 to get the sequence \(2^1, 2^2, 2^3, 2^4\). The sum of this sequence is \(2 + 4 + 8 + 16 = 30\). So, \(\sum_{j=1}^{4} 2^{j} = 30\).
02

Calculate the right side sum

We calculate the sum on the right side of the equation. Here the sequence is slightly more involved because of the term \(j-2\) in the exponent. We substitute in \(j = 3, 4, 5, 6\) to get \(2^{3-2}, 2^{4-2}, 2^{5-2}, 2^{6-2}\), which gives the sequence \(2, 4, 8, 16\). The sum of this sequence also equals \(30\). So, \(\sum_{j=3}^{6} 2^{j-2} = 30\).
03

Compare the results

After calculating the sums on both sides of the equation, we find that they both equal 30. Therefore, \(\sum_{j=1}^{4} 2^{j} = \sum_{j=3}^{6} 2^{j-2}\) is a true statement since both sums equal to 30.

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