Questions: Evaluate the following expressions by two methods. (i) Use Theorem 5.1. (ii) Use a calculator. a. ∑ from k=1 to 45 of k b. ∑ from k=1 to 45 of (5k - 1) c. ∑ from k=1 to 75 of 2k² d. ∑ from n=1 to 50 of (1 + n²) e. ∑ from m=1 to 75 of (2m + 2)/3 f. ∑ from j=1 to 20 of (3j - 4) g. ∑ from p=1 to 35 of (2p + p²) h. ∑ from n=0 to 40 of (n² + 3n - 1)

Evaluate the sum \(\sum_{k=1}^{45} k\).

Using Theorem 5.1

The formula for the sum of the first \(n\) natural numbers is \(\sum_{k=1}^{n} k = \frac{n(n+1)}{2}\). Applying this formula for \(n = 45\), we get:

\[ \sum_{k=1}^{45} k = \frac{45(45+1)}{2} = \frac{45 \cdot 46}{2} = 45 \cdot 23 = 1035 \]

Using a calculator

Using Python or a calculator, compute the sum directly:

python sum_k = sum(range(1, 46)) print(sum_k) # Output: 1035

\\(\\boxed{1035}\\)


Evaluate the sum \\(\\sum_{k=1}^{45}(5 k-1)\\).

Using Theorem 5.1

Break down the sum using properties of summation:

\\[
\\sum_{k=1}^{45}(5k - 1) = 5\\sum_{k=1}^{45} k - \\sum_{k=1}^{45} 1
\\]

Calculate each part:

\\[
5\\sum_{k=1}^{45} k = 5 \\cdot 1035 = 5175
\\]

\\[
\\sum_{k=1}^{45} 1 = 45
\\]

Combine the results:

\\[
5175 - 45 = 5130
\\]

Using a calculator

Using Python or a calculator, compute the sum directly:

python
sum_k = sum(5*k - 1 for k in range(1, 46))
print(sum_k)  # Output: 5130

\(\boxed{5130}\)

Evaluate the sum \(\sum_{k=1}^{75} 2 k^{2}\).

Using Theorem 5.1

Use the formula for the sum of squares:

\[ \sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6} \]

Apply the formula for \(n = 75\):

\[ 2 \sum_{k=1}^{75} k^2 = 2 \cdot \frac{75(76)(151)}{6} = \frac{75 \cdot 76 \cdot 151}{3} = 25 \cdot 76 \cdot 151 = 286900 \]

Using a calculator

Using Python or a calculator, compute the sum directly:

python sum_k = sum(2*k**2 for k in range(1, 76)) print(sum_k) # Output: 286900

\\(\\boxed{286900}\\)


Evaluate the sum \\(\\sum_{n=1}^{50}\\left(1+n^{2}\\right)\\).

Using Theorem 5.1

Break down the sum:

\\[
\\sum_{n=1}^{50} (1 + n^2) = \\sum_{n=1}^{50} 1 + \\sum_{n=1}^{50} n^2
\\]

Calculate each part:

\\[
\\sum_{n=1}^{50} 1 = 50
\\]

\\[
\\sum_{n=1}^{50} n^2 = \\frac{50(51)(101)}{6} = 42925
\\]

Combine the results:

\\[
50 + 42925 = 42975
\\]

Using a calculator

Using Python or a calculator, compute the sum directly:

python
sum_n = sum(1 + n**2 for n in range(1, 51))
print(sum_n)  # Output: 42975

\(\boxed{42975}\)

\(\boxed{1035}\)
\(\boxed{5130}\)
\(\boxed{286900}\)
\(\boxed{42975}\)