Problem 125
Question
Determine whether the statement is true or false. Justify your answer. $$\sum_{i=1}^{4}\left(i^{2}+2 i\right)=\sum_{i=1}^{4} i^{2}+2 \sum_{i=1}^{4} i$$
Step-by-Step Solution
Verified Answer
The statement is true.
1Step 1: Calculate left-hand side
Start by evaluating the left-hand side of the equation, which is the sum of \((i^{2}+2i)\) from i=1 to 4. Iterate over each integer from 1 to 4, plug it into the expression, and then add up the resulting values: \((1^2+2*1)+(2^2+2*2)+(3^2+2*3)+(4^2+2*4) = 3+8+15+24 = 50.
2Step 2: Calculate right-hand side
Next, calculate the right-hand side of the equation, which is given as the sum of \(i^{2}\) from i=1 to 4 plus twice the sum of \(i\) from i=1 to 4. As in the previous step, iterate over each integer from 1 to 4, plug it into both expressions, and then add up the resulting values: \((1^2+2^2+3^2+4^2) + 2*(1+2+3+4) = 30 + 20 = 50.
3Step 3: Compare both sides
Now that the values on both sides are computed, compare them to determine whether the original statement is true or false. Since the calculated result is 50 for both sides, the statement is true.
Other exercises in this chapter
Problem 124
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The sum of the numbers in the \(n\) th row of Pascal's Triangle is \(2^{n}\).
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Determine whether the statement is true or false. Justify your answer. $$\sum_{j=1}^{4} 2^{j}=\sum_{j=3}^{6} 2^{j-2}$$
View solution