Problem 126 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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, \(\sum_{j=1}^{4} 2^{j} = 30\).

Step 2: 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, \(\sum_{j=3}^{6} 2^{j-2} = 30\).

Step 3: 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.