Problem 58 - Precalculus with Limits

Question

In Exercises 49 - 58, find the sum using the formulas for the sums of powers of integers. \( \sum_{j=1}^{10}\left(3 - \dfrac{1}{2}j + \dfrac{1}{2}j^2\right) \)

Step-by-Step Solution

Answer

The sum of the sequence from 1 to 10 of the function \(3 - \dfrac{1}{2}j + \dfrac{1}{2}j^2\) is 101.25.

Step 1: Sum of the constant sequence

The sum of a constant sequence can be calculated by multiplying the constant (3) by the number of terms (10). So, \(sum_1 = 3 * 10 = 30\).

Step 2: Find the sum of the linear sequence

The sum of a linear sequence \(-\dfrac{1}{2}j\) from 1 to n can be calculated with the formula \(\dfrac{n^2}{2}\), so \(sum_2 = -\dfrac{1}{2} * \dfrac{10^2}{2} = -25\).

Step 3: Calculate the sum of quadratic sequence

The sum of a quadratic sequence \(\dfrac{1}{2}j^2\) from 1 to n can be calculated using the formula \(\dfrac{n(n+1)(2n+1)}{12}\). With this formula, \(sum_3 = \dfrac{1}{2}*\dfrac{10 * 11 * 21}{12} = 96.25\).

Step 4: Add up all individual sums

Now, the final result is obtained by adding all the individual sums: 30 - 25 + 96.25 = 101.25.

Key Concepts

Sum of Powers

Summing powers is a handy way to add up numbers raised to a certain power across a range of values. To break this down, let’s consider the sequence given in the exercise: \The sequence is expressed as:

• Constant: 3
• Linear: \(-\frac{1}{2}j\)
• Quadratic: \(\frac{1}{2}j^2\)

Summing these powers involves breaking down the entire equation into individual components. Each is treated independently, and then summed up at the end.
This makes use of known formulas for summing different kinds of power sequences, ensuring a systematic and accurate solution.
This approach is especially useful with more complex sequences, as it keeps everything organized and easy to follow.

Linear Sequence Sum

Linear sequences are sequences where the difference between each consecutive term is constant. For instance, the sequence\(-\frac{1}{2}, -1, -\frac{3}{2}, \ldots\) results from \(-\frac{1}{2}j\) for each step. The formula used to sum these linear sequences up to \(n\) terms is: \\[\sum_{j=1}^{n} j = \frac{n(n+1)}{2} \]
In our exercise, a slight modification applies because of the coefficient of \(-\frac{1}{2}\). This turns the formula into \\[ \sum_{j=1}^{n} -\frac{1}{2}j = -\frac{1}{2} \times \frac{n(n+1)}{2} \]
For \(n=10\), this equals \\(-\frac{1}{2} \times 25 = -25\).
This exemplifies how changing the sequence’s coefficient affects the outcome, making comprehending linear summation invaluable for problem-solving.

Quadratic Sequence Sum

Quadratic sequences involve squaring each term in the sequence, and these terms increase more quickly compared to linear sequences. The sum formula for quadratic sequences is:\[ \sum_{j=1}^{n} j^2 = \frac{n(n+1)(2n+1)}{6} \]
The exercise problem included a sequence that involved \(\frac{1}{2}j^2\) — indicating the need to multiply the normal formula result by \(\frac{1}{2}\).
For \(n=10\), the calculation is:

• Calculate \(10 \times 11 \times 21 = 2310\)
• Divide by \(12\) to get \(192.5\) (one-sixth of the previous result due to the standard formula)
• Finally, apply the \(\frac{1}{2}\) factor for the actual expression in the problem, reaching \(96.25\)

This process shows how to adapt the standard quadratic sum formula accurately, emphasizing the role of coefficients in determining the sum of sequences. Understanding these adaptations can reveal more about diverse series' behavior in math.