Problem 34 - Algebra and Trigonometry with Analytic Geometry

Question

Express the sum in terms of \(n\). $$ \sum_{k=1}^{n}\left(3 k^{2}-2 k+1\right) $$

Step-by-Step Solution

Answer

\( \sum_{k=1}^{n} (3k^2-2k+1) = \frac{2n^3+n^2+n}{2} \).

Step 1: Break Apart the Sum

To express the sum \( \sum_{k=1}^{n} (3k^2 - 2k + 1) \), we will break it into three separate sums: \( 3\sum_{k=1}^{n} k^2 - 2\sum_{k=1}^{n} k + \sum_{k=1}^{n} 1 \). This allows us to handle each part of the expression individually.

Step 2: Evaluate the Sum of \( k^2 \)

The formula for the sum of squares from 1 to \( n \) is \( \sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6} \). Thus, \( 3\sum_{k=1}^{n} k^2 = 3\times\frac{n(n+1)(2n+1)}{6} = \frac{n(n+1)(2n+1)}{2} \).

Step 3: Evaluate the Sum of \( k \)

The formula for the sum of integers from 1 to \( n \) is \( \sum_{k=1}^{n} k = \frac{n(n+1)}{2} \). Thus, \( -2\sum_{k=1}^{n} k = -2\times\frac{n(n+1)}{2} = -n(n+1) \).

Step 4: Evaluate the Sum of 1

The sum of 1 \( n \) times is simply \( \sum_{k=1}^{n} 1 = n \).

Step 5: Combine the Sums

Combine the results from Steps 2, 3, and 4: \[ \frac{n(n+1)(2n+1)}{2} - n(n+1) + n \]. Simplify this expression by carefully distributing and combining like terms.

Step 6: Simplify the Expression

Start by expanding the terms and simplifying:\[ \frac{n(n+1)(2n+1)}{2} - n(n+1) + n = \frac{2n^3 + 3n^2 + n}{2} - n^2 - n + n \].Simplify further to obtain: \[ \frac{2n^3 + 3n^2 + n}{2} - n^2 \]. Finally, simplify completely to \[ \frac{2n^3 + n^2 + n}{2} \].

Key Concepts

Sum of Squares Formula

The sum of squares formula is a powerful mathematical tool that helps you calculate the sum of the squares of the first \( n \) integers.
This formula is quite handy when dealing with series where the squares of numbers appear frequently. To express mathematically, the formula is:\[\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}\]This formula facilitates the quick computation of the sum by substituting the desired number \( n \) into the formula and obtaining the result.
For example, if you want to find the sum of squares of numbers from 1 to 5, the formula will help you do this in just a few steps:- Set \( n = 5 \) in the formula.- Calculate \( \frac{5(5+1)(2\times5+1)}{6} \).- Simplify it to get the result.The use of the sum of squares formula is essential in simplifying complex expressions involving squares, as it turns a lengthy summation process into a straightforward computation.

Sum of Integers Formula

The sum of integers formula is an essential algebraic expression used to find the sum of a succession of numbers from 1 to \( n \).
This formula is simple yet very effective for adding a series of consecutive integers.The formula is given by:\[\sum_{k=1}^{n} k = \frac{n(n+1)}{2}\]This means you just have to plug in the value of \( n \) to quickly sum the numbers from 1 up to \( n \).
It simplifies tasks dramatically compared to adding each integer one at a time.
Let's observe a quick example:- Consider \( n = 4 \).- Plug into the formula: \( \frac{4(4+1)}{2} \).- Solve to find the sum equals 10.Using the sum of integers formula is a time saver, especially when dealing with large numbers or when you need to perform algebraic manipulations that require summing series of integers.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying algebraic expressions to make them easier to understand or solve.
It includes a variety of techniques such as combining like terms, expanding expressions, and factoring.In the context of summations, especially when dealing with more complex terms like \( 3k^2 - 2k + 1 \), algebraic manipulation allows us to break down the expression into simpler parts.
Here's how it can be applied:- **Break Down**: Separate the combined expression into distinct sums.- **Simplify Each Part**: Use known formulas like the sum of squares and sum of integers to handle each portion separately.- **Combine and Simplify**: Once each part is calculated, recombine the results and simplify the final equation by combining like terms.For instance, given the summation \( \sum_{k=1}^{n}(3k^2 - 2k + 1) \), you can split it into separate sums for \( k^2 \), \( k \), and the constant term.
Simplifying each individually before bringing them together results in a cleaner, more understandable solution.
It allows you to better identify and manage each component of the algebraic expression. Algebraic manipulation gives you the tools to dissect and reconstruct complex mathematical problems effectively.