Comprehensive Calculus II Study Guide – Step-by-Step Guidance

Q1. Write the following sum in summation notation then evaluate: 12 + 22 + 32 + · · · + 2292 + 2302 + 2312

Background

Topic: Summation Notation and Finite Sums

This question tests your ability to express a sequence as a sum using sigma notation and then evaluate the sum, likely using formulas for the sum of squares.

Key Terms and Formulas

• Summation notation: $\sum_{i=1}^{n} a_i$

• Sum of squares formula: $\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}$

Step-by-Step Guidance

1. Identify the pattern in the sequence. Notice that the terms are consecutive squares: $12^2, 22^2, 32^2, \ldots, 231^2$.

2. Express the sum in sigma notation. Determine the starting and ending indices for $i$.

3. Write the sum as $\sum_{i=a}^{b} i^2$, where $a$ and $b$ are the first and last numbers in the sequence.

4. Recall the formula for the sum of squares and set up the calculation for the given range.

Try solving on your own before revealing the answer!

Q2. Evaluate $\sum_{i=1}^{n} i(i + 1)$

Background

Topic: Summation Techniques

This question asks you to evaluate a sum involving a quadratic expression in $i$. You may need to expand the expression and use known summation formulas.

Key Terms and Formulas

• Summation properties: $\sum_{i=1}^{n} (a_i + b_i) = \sum_{i=1}^{n} a_i + \sum_{i=1}^{n} b_i$

• Sum of first $n$ integers: $\sum_{i=1}^{n} i = \frac{n(n+1)}{2}$

• Sum of first $n$ squares: $\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}$

Step-by-Step Guidance

1. Expand $i(i+1)$ to $i^2 + i$ inside the summation.

2. Split the sum: $\sum_{i=1}^{n} i^2 + \sum_{i=1}^{n} i$.

3. Apply the formulas for $\sum_{i=1}^{n} i^2$ and $\sum_{i=1}^{n} i$.

4. Combine the results to get a single expression in terms of $n$.

Try solving on your own before revealing the answer!

Q3. Evaluate $\sum_{i=3}^{50} i^3$

Background

Topic: Summation of Powers

This question asks you to compute the sum of cubes from $i=3$ to $i=50$. You will use the formula for the sum of cubes and adjust for the lower limit.

Key Terms and Formulas

• Sum of cubes: $\sum_{i=1}^{n} i^3 = \left(\frac{n(n+1)}{2}\right)^2$

Step-by-Step Guidance

1. Recall that $\sum_{i=3}^{50} i^3 = \sum_{i=1}^{50} i^3 - \sum_{i=1}^{2} i^3$.

2. Compute $\sum_{i=1}^{50} i^3$ using the sum of cubes formula.

3. Compute $\sum_{i=1}^{2} i^3$ using the same formula.

4. Subtract the two results to get the sum from $i=3$ to $i=50$.

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Q4. Evaluate $\sum_{i=n}^{2n} i^3$

Background

Topic: Summation of Powers with Variable Limits

This question asks you to sum cubes from $i=n$ to $i=2n$. You will use the sum of cubes formula and adjust for the lower limit.

Key Terms and Formulas

• Sum of cubes: $\sum_{i=1}^{m} i^3 = \left(\frac{m(m+1)}{2}\right)^2$

Step-by-Step Guidance

1. Express $\sum_{i=n}^{2n} i^3$ as $\sum_{i=1}^{2n} i^3 - \sum_{i=1}^{n-1} i^3$.

2. Apply the sum of cubes formula to both $2n$ and $n-1$.

3. Write out the expressions for each sum.

4. Subtract the two results to get the sum from $i=n$ to $i=2n$.

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Q5. Evaluate $\sum_{i=3}^{n} (2i^2 - i)$

Background

Topic: Linear Combination of Summations

This question asks you to evaluate a sum involving both $i^2$ and $i$ terms, starting from $i=3$ to $i=n$.

Key Terms and Formulas

• Sum of squares: $\sum_{i=1}^{m} i^2 = \frac{m(m+1)(2m+1)}{6}$

• Sum of first $m$ integers: $\sum_{i=1}^{m} i = \frac{m(m+1)}{2}$

Step-by-Step Guidance

1. Split the sum: $2\sum_{i=3}^{n} i^2 - \sum_{i=3}^{n} i$.

2. Express each sum as the difference between the sum from $i=1$ to $n$ and $i=1$ to $2$.

3. Apply the formulas for $\sum i^2$ and $\sum i$ to each part.

4. Combine the results to get a single expression in terms of $n$.

Try solving on your own before revealing the answer!

Q6. Evaluate $\sum_{i=3}^{50} (2i)$

Background

Topic: Summation of Linear Functions

This question asks you to sum a linear function of $i$ from $i=3$ to $i=50$.

Key Terms and Formulas

• Sum of first $m$ integers: $\sum_{i=1}^{m} i = \frac{m(m+1)}{2}$

Step-by-Step Guidance

1. Factor out the constant: $2\sum_{i=3}^{50} i$.

2. Express $\sum_{i=3}^{50} i$ as $\sum_{i=1}^{50} i - \sum_{i=1}^{2} i$.

3. Apply the formula for the sum of the first $m$ integers to both terms.

4. Multiply the result by 2 to get the final sum.

Try solving on your own before revealing the answer!