Problem 4 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the value of the indicated sum. $$ \sum_{l=3}^{8}(l+1)^{2} $$

Step-by-Step Solution

Answer

The sum is 271.

Step 1: Understand the Summation

The summation notation \( \sum_{l=3}^{8}(l+1)^{2} \) indicates that we need to find the sum of the square of \((l+1)\) for each integer \( l \) from 3 to 8, inclusive. This means we will calculate \((l+1)^2\) for each \( l \) in this range and then sum those results.

Step 2: Calculate Each Term

Calculate \((l+1)^2\) for each value of \( l \) from 3 to 8.- For \( l = 3 \), \((3+1)^2 = 4^2 = 16\).- For \( l = 4 \), \((4+1)^2 = 5^2 = 25\).- For \( l = 5 \), \((5+1)^2 = 6^2 = 36\).- For \( l = 6 \), \((6+1)^2 = 7^2 = 49\).- For \( l = 7 \), \((7+1)^2 = 8^2 = 64\).- For \( l = 8 \), \((8+1)^2 = 9^2 = 81\).

Step 3: Sum the Calculated Terms

Now add up all the calculated terms: \(16 + 25 + 36 + 49 + 64 + 81\).

Step 4: Perform the Addition

Add the numbers step by step:- \( 16 + 25 = 41 \)- \( 41 + 36 = 77 \)- \( 77 + 49 = 126 \)- \( 126 + 64 = 190 \)- \( 190 + 81 = 271 \)

Step 5: Conclusion

The sum of \( \sum_{l=3}^{8}(l+1)^2 \) is 271.

Key Concepts

Step by Step Problem Solving

Breaking down mathematical problems into smaller, manageable steps can transform a complex problem into an understandable task. With the exercise of finding the sum \( \sum_{l=3}^{8}(l+1)^{2} \), we approached the problem by understanding each component of the summation.

• Begin by identifying the values of \( l \) in the range, from \( 3 \) to \( 8 \).
• For each \( l \), determine \((l+1)^2\), calculating the squared value for each term.
• Sum these individual results to find the final answer.

It's crucial to thoroughly understand each step, ensuring clarity with each calculation and not moving on until sure of the result. This methodology aids in avoiding mistakes and builds a strong foundation for tackling future tasks.

Mathematical Notation

Mathematical notation is like a language that allows us to convey complex ideas succinctly. In this exercise, the notation \( \sum_{l=3}^{8}(l+1)^2 \) tells us several things:

• The \( \sum \) symbol, indicates summation. It sums up a sequence of numbers defined by the expression following it.
• The index of summation \( l \) starts at \( 3 \) and ends at \( 8 \), these are the limits of the summation range.
• The expression \((l+1)^2\) shows the operation we'll perform on each value of \( l \).

Understanding these symbols and their meaning is key to unlocking the potential of concise mathematical communication. This will also enable tackling more complex calculations effectively, as they form the basis of many mathematical tasks.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and arithmetic operations. In our exercise, \((l+1)^2\) is the algebraic expression used. Breaking it down, this expression can be understood as follows:

• The term \( l+1 \) is an equation that increases each value of \( l \) by \( 1 \).
• Squaring \( (l+1) \) changes how we treat each result, indicating each sum will get squared.

The use of algebraic expressions helps in formulating problems compactly, letting us easily manipulate and simplify calculations. Mastery of them allows smoother problem-solving across mathematics.

Polynomials

Though the exercise does not deal explicitly with polynomials, understanding them enriches your mathematical toolkit. A polynomial is an algebraic expression consisting of variables raised to whole number powers. The term \( (l+1)^2 \) within our exercise is a simple polynomial:

• As a polynomial, \((l+1)^2\) can be expanded to \( l^2 + 2l + 1 \).
• The polynomial can represent a curve in geometry or physics, showing how its roots, coefficients, and power affect its graph.

Polynomials appear in many mathematical problems and real-world applications. Understanding their properties and behaviors is integral for further studies in calculus and algebra.