$$ \begin{array}{l} ext { Finding a Partial Sum } \ ext { partial sum. }\end{array} $$ $$ \begin{array}{ll} ext { 59. } \sum_{n=1}^{50} n & ext { 60. } \sum_{n=51}^{100} 7 n \ ext { 61. } \sum_{n=11}^{30} n-\sum_{n=1}^{10} n & ext { 62. } \sum_{n=51}^{100} n-\sum_{n=1}^{50} n\end{array} $$

Question

$$ \begin{array}{l}	ext { Finding a Partial Sum } \ 	ext { partial sum. }\end{array} $$ $$ \begin{array}{ll}	ext { 59. } \sum_{n=1}^{50} n & 	ext { 60. } \sum_{n=51}^{100} 7 n \ 	ext { 61. } \sum_{n=11}^{30} n-\sum_{n=1}^{10} n & 	ext { 62. } \sum_{n=51}^{100} n-\sum_{n=1}^{50} n\end{array} $$

UpStudy AI · Accepted Answer

**59. $$\sum_{n=1}^{50} n$$**

We use the formula for the sum of the first $$ N $$ natural numbers:
$$
\sum_{n=1}^{N} n = \frac{N(N+1)}{2}.
$$
For $$ N = 50 $$:
$$
\sum_{n=1}^{50} n = \frac{50(51)}{2} = \frac{2550}{2} = 1275.
$$

---

**60. $$\sum_{n=51}^{100} 7n$$**

Factor out the constant 7:
$$
\sum_{n=51}^{100} 7n = 7 \sum_{n=51}^{100} n.
$$
The sum from $$ n=51 $$ to $$ n=100 $$ can be written as:
$$
\sum_{n=51}^{100} n = \sum_{n=1}^{100} n - \sum_{n=1}^{50} n.
$$
Calculating each part:
$$
\sum_{n=1}^{100} n = \frac{100(101)}{2} = 5050,
$$
$$
\sum_{n=1}^{50} n = 1275 \quad \text{(from part 59)}.
$$
Thus,
$$
\sum_{n=51}^{100} n = 5050 - 1275 = 3775.
$$
Multiplying by 7:
$$
\sum_{n=51}^{100} 7n = 7 \times 3775 = 26425.
$$

---

**61. $$\sum_{n=11}^{30} n - \sum_{n=1}^{10} n$$**

First, we compute $$\sum_{n=11}^{30} n$$. There are $$30 - 11 + 1 = 20$$ terms, and the sum of an arithmetic series is given by:
$$
\text{Sum} = \frac{\text{number of terms} \times (\text{first term} + \text{last term})}{2}.
$$
Thus,
$$
\sum_{n=11}^{30} n = \frac{20(11+30)}{2} = \frac{20 \times 41}{2} = 10 \times 41 = 410.
$$
Next, compute $$\sum_{n=1}^{10} n$$:
$$
\sum_{n=1}^{10} n = \frac{10(11)}{2} = 55.
$$
Subtracting:
$$
\sum_{n=11}^{30} n - \sum_{n=1}^{10} n = 410 - 55 = 355.
$$

---

**62. $$\sum_{n=51}^{100} n - \sum_{n=1}^{50} n$$**

We already computed:
$$
\sum_{n=1}^{100} n = 5050 \quad \text{and} \quad \sum_{n=1}^{50} n = 1275.
$$
The sum from $$ n=51 $$ to $$ n=100 $$:
$$
\sum_{n=51}^{100} n = 5050 - 1275 = 3775.
$$
Then, subtract $$\sum_{n=1}^{50} n$$ from this:
$$
\sum_{n=51}^{100} n - \sum_{n=1}^{50} n = 3775 - 1275 = 2500.
$$

---

Final Answers:

- **59.** $$1275$$
- **60.** $$26425$$
- **61.** $$355$$
- **62.** $$2500$$
**59.** 1275 **60.** 26425 **61.** 355 **62.** 2500

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