4. Calculate; $$ \sum_{k=1}^{30}(3 k+5) $$ 5. Evaluate: $$ \sum_{t=0}^{99}(3 t-1) $$

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UpStudy AI · Accepted Answer

Let's evaluate each of the given summations step by step.

---

### **4. Calculate: $$ \sum_{k=1}^{30}(3k + 5) $$**

**Step 1: Break Down the Summation**

The summation can be separated into two parts:

$$
\sum_{k=1}^{30}(3k + 5) = 3 \sum_{k=1}^{30}k + \sum_{k=1}^{30}5
$$

**Step 2: Compute Each Part Individually**

1. **Sum of $$ k $$ from 1 to 30:**

$$
\sum_{k=1}^{30}k = \frac{n(n + 1)}{2} = \frac{30 \times 31}{2} = 465
$$

2. **Sum of constants (5) over 30 terms:**

$$
\sum_{k=1}^{30}5 = 5 \times 30 = 150
$$

**Step 3: Combine the Results**

$$
3 \sum_{k=1}^{30}k + \sum_{k=1}^{30}5 = 3 \times 465 + 150 = 1395 + 150 = 1545
$$

**Final Answer:**

$$
\sum_{k=1}^{30}(3k + 5) = **1,\!545**
$$

---

### **5. Evaluate: $$ \sum_{t=0}^{99}(3t - 1) $$**

**Step 1: Break Down the Summation**

The summation can be separated into two parts:

$$
\sum_{t=0}^{99}(3t - 1) = 3 \sum_{t=0}^{99}t - \sum_{t=0}^{99}1
$$

**Step 2: Compute Each Part Individually**

1. **Sum of $$ t $$ from 0 to 99:**

$$
\sum_{t=0}^{99}t = \frac{n(n + 1)}{2} = \frac{99 \times 100}{2} = 4,\!950
$$

2. **Sum of constants (-1) over 100 terms (since $$ t $$ goes from 0 to 99):**

$$
\sum_{t=0}^{99}1 = 1 \times 100 = 100
$$

**Step 3: Combine the Results**

$$
3 \sum_{t=0}^{99}t - \sum_{t=0}^{99}1 = 3 \times 4,\!950 - 100 = 14,\!850 - 100 = 14,\!750
$$

**Final Answer:**

$$
\sum_{t=0}^{99}(3t - 1) = **14,\!750**
$$

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If you have any further questions or need additional explanations, feel free to ask!
The sum $$ \sum_{k=1}^{30}(3k + 5) $$ equals **1,545**. The sum $$ \sum_{t=0}^{99}(3t - 1) $$ equals **14,750**.

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