Question

Find the sum of each sequence. \(\sum_{k=1}^{16}\left(k^{2}+4\right)\)

Short answer

1560

Step-by-step solution

Understand the Summation Notation

The notation \(n \sum_{k=1}^{16}(k^2 + 4)\) means to sum the expression \(n (k^2 + 4)n\) for each integer \(n kn\) from \(n 1 n\) to \(n 16n\).

Separate the Summation

The summation of a sum can be separated into individual summations: \(n \sum_{k=1}^{16} (k^2 + 4) = \sum_{k=1}^{16} k^2 + \sum_{k=1}^{16} 4 n\).

Find the Summation of Constant

The sum of a constant \(n 4n\) taken \(n 16n\) times is \(n 4 \times 16 = 64n\).

Find the Summation of Squares

Use the formula for the sum of the squares of the first \(n n n\) positive integers: \( \sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6} n \). For \( n = 16 n\), this becomes: \(n \sum_{k=1}^{16} k^2 = \frac{16 \times 17 \times 33}{6} = 1496 n\).

Combine Both Summations

Finally, add the results from Step 3 and Step 4: \(n 1496 + 64 = 1560 n\).

Key concepts

Summation Notation

Summation notation is a convenient way to represent the sum of a sequence of terms. In the notation \(\sum_{k=1}^{16}(k^2 + 4)\), the Greek letter sigma (\(\sum\)) is used to represent summation. The expression\(\sum_{k=1}^{16}\) tells us to sum the terms from \(k=1\) to \(k=16\) of the expression \(k^2 + 4\). Each term in the sequence is generated by substituting the integer values of \(k\) (from 1 to 16) into \(k^2 + 4\). The resulting terms are then added together.

Sum of Squares Formula

The sum of squares formula is a mathematical formula used to find the sum of the squares of the first \(n\) positive integers. The formula is expressed as: \[\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}\]. This formula helps in quickly calculating the sum without having to individually square each number and add them. For example, to find the sum of the squares of the first 16 positive integers, we substitute \(n = 16\) into the formula: \[ \sum_{k=1}^{16} k^2 = \frac{16 \times 17 \times 33}{6} = 1496 \].

Separating Summations

Separating summations is a useful technique that simplifies the summation process by breaking down a complex sum into simpler parts. This is particularly useful when dealing with a sum of multiple terms or expressions. For instance, in the expression \( \sum_{k=1}^{16}(k^2 + 4)\), we can separate it into two individual summations: \( \sum_{k=1}^{16} k^2 + \sum_{k=1}^{16} 4 \). Each of these summations can then be handled separately. This method leverages the linearity of summation, making it easier to calculate the overall sum.

Constant Summation

Constant summation refers to the process of summing a constant value over a specified range. When the expression inside the sum is a constant, such as in \( \sum_{k=1}^{16} 4 \), the sum can be calculated by multiplying the constant by the number of terms. In this example, the constant \(4\) is summed from \(k=1\) to \(k=16\), yielding \(4 \times 16 = 64\). This method simplifies the summation of constants and is a quick way to handle such sums.