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Magazine Chapter 8: Problem 71
Find the sum for each series. $$\sum_{i=1}^{77}\left(i^{2}+52 i+672\right)$$
Short Answer
Expert verified
The sum of the series is calculated by adding the sums from each component: squares, linear, and constant expressions.
Step by step solution
01
Understand the Problem
We need to find the sum of a series where each term of the series is given by the expression \(i^2 + 52i + 672\), with \(i\) ranging from 1 to 77. This means we need to compute \(\sum_{i=1}^{77} \left(i^2 + 52i + 672\right)\).
02
Break Down the Series
To simplify the computation, we can split the series into three separate sums: \(\sum_{i=1}^{77} i^2\), \(\sum_{i=1}^{77} 52i\), and \(\sum_{i=1}^{77} 672\). These are the sum of squares, the sum of a linear sequence, and the sum of a constant, respectively.
03
Calculate the Sum of Squares
The formula for the sum of the squares of the first \(n\) natural numbers is \(\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}\). For \(n = 77\), the sum is \(\sum_{i=1}^{77} i^2 = \frac{77 \cdot 78 \cdot 155}{6}\). Calculate this to find the sum of the squares.
04
Calculate the Sum of the Linear Sequence
The sum of the linear sequence \(\sum_{i=1}^{n} i\) is given by the formula \(\frac{n(n+1)}{2}\). With a coefficient of 52, the sum for \(i = 1\) to \(77\) becomes \(52 \sum_{i=1}^{77} i = 52 \cdot \frac{77 \cdot 78}{2}\). Compute this to find the linear sum.
05
Calculate the Sum of the Constant
The sum of the constant sequence \(\sum_{i=1}^{n} c\) is simply \(c \cdot n\), where \(c\) is the constant. In this case, \(672 \times 77 = 51864\). Calculate this value.
06
Add the Results
Add all calculated sums: the sum from squares, the sum from the linear sequence, and the sum of the constant. Combine these results to get the total sum of the series.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Sum of Squares
In mathematics, the sum of squares is a common way to understand how numbers add up in quadratic terms. It involves summing up each number's square from 1 up to a certain number, in this case, 77. This series can be quite large, and we use a specific formula to make calculations easier:
- \( \sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6} \)
Linear Sequence
A linear sequence involves numbers that increase or decrease by a constant amount. In our exercise, each term includes 52 times a linearly increasing number (\(i\)). The sum of a linear sequence follows a straightforward formula:
- \( \sum_{i=1}^{n} i = \frac{n(n+1)}{2} \)
Constant Series
When dealing with a constant series, the goal is straightforward: each term of the series remains the same. For a constant value like 672 repeated over 77 terms, we simply multiply the constant by the number of terms. The formula here is:
- \( \sum_{i=1}^{n} c = c \cdot n \)
Algebraic Formula
Algebraic formulas are crucial for breaking down and solving complex mathematical problems like our series sum exercise. They allow us to take a complicated expression and find a total sum efficiently. By using algebraic formulas:
- We simplify calculations, breaking complex expressions into simpler parts.
- We apply specific formulas like those for sum of squares or linear sequences.
- These formulas are derived from fundamental principles of algebra.
