Problem 4 - Calculus

Question

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

Step-by-Step Solution

Answer

The value of the sum is 271.

Step 1: Understand the Expression

The given expression is a summation: \(\sum_{l=3}^{8}(l+1)^{2}\). This means we need to sum up values obtained by squaring \(l+1\) for each integer value of \(l\) from 3 to 8 inclusive.

Step 2: Expand and Evaluate the Expression Inside the Summation

First rewrite the expression inside as \((l+1)^{2} = l^2 + 2l + 1\). We will evaluate this for each value of \(l\) from 3 to 8.

Step 3: Calculate Individual Terms

Calculate \((l+1)^{2}\) for each integer \(l\) from 3 to 8:- For \(l=3\), \((3+1)^2 = 16\)- For \(l=4\), \((4+1)^2 = 25\)- For \(l=5\), \((5+1)^2 = 36\)- For \(l=6\), \((6+1)^2 = 49\)- For \(l=7\), \((7+1)^2 = 64\)- For \(l=8\), \((8+1)^2 = 81\)

Step 4: Sum the Results

Add all the individual terms calculated:\(16 + 25 + 36 + 49 + 64 + 81\).

Step 5: Compute the Final Sum

Perform the addition:\(16 + 25 + 36 + 49 + 64 + 81 = 271\).

Key Concepts

Summation notation

Summation notation is a way to represent the sum of a sequence of terms. It's denoted by the Greek capital letter Sigma (\( \Sigma \)), and it typically includes a formula to calculate each term in the sequence. Let's break down the notation used in this exercise:

• The notation \( \sum_{l=3}^{8} \) indicates we are summing from \( l = 3 \) to \( l = 8 \).
• Inside the summation, \( (l+1)^{2} \) is the expression you evaluate at each integer \( l \) within the range.

Summation notation makes it easier to work with large and complex expressions by compactly representing sums over large sets of numbers. In this exercise, it allows us to calculate the sum of squares of numbers incremented by 1, from \( l = 3 \) to \( l = 8 \).

Polynomial expansion

The process of polynomial expansion involves rewriting expressions in an expanded form.
In our exercise, the term \( (l+1)^{2} \) is expanded as a polynomial:

• First, rewrite the squared term: \( (l+1)^{2} = l^2 + 2l + 1 \).
• This is a basic example of a binomial expansion, where the expression is expanded into a sum of terms.

By expanding \( (l+1)^{2} \), it becomes easier to understand and evaluate, as each part of the expression can be computed separately. This helps simplify the calculation process and aids in identifying patterns or properties within the expression, such as symmetry or cancellations.

Step-by-step calculation

Breaking problems down into smaller, more manageable steps is a useful strategy in mathematics. This is particularly effective for complex summations, where each step builds on the last to reach a final solution.
Here's the approach used in this exercise:

• Identify the expression and rewrite it in a form that’s easy to calculate for each term \( (l+1)^{2} \to l^2 + 2l + 1 \).
• Calculate the expanded polynomial for each specific value of \( l \) in the summation range from 3 to 8.
• Sum all the individual results to find the total sum, checking each step for accuracy.
  Summing term by term helps in understanding each part’s contribution to the overall result.

This step-by-step approach ensures the problem is tackled in an organized manner, making the solution easier to understand and verify.