$$ \sum _ { t = - 2 } ^ { 30 } ( 4 t + 6 ) $$

Question

UpStudy AI · Accepted Answer

To evaluate the summation:

$$
\sum_{t=-2}^{30} (4t + 6)
$$

we can break it down into two separate sums:

$$
\sum_{t=-2}^{30} 4t + \sum_{t=-2}^{30} 6
$$

Let's compute each part step by step.

### 1. Calculate $$\sum_{t=-2}^{30} 4t$$

Factor out the constant $$4$$:

$$
4 \sum_{t=-2}^{30} t
$$

First, determine the number of terms in the summation:

$$
\text{Number of terms} = 30 - (-2) + 1 = 33
$$

Now, find the sum of an arithmetic series using the formula:

$$
\sum_{t=a}^{b} t = \frac{n}{2} (a + b)
$$

Where:
- $$n = 33$$ (number of terms)
- $$a = -2$$ (first term)
- $$b = 30$$ (last term)

Plugging in the values:

$$
\sum_{t=-2}^{30} t = \frac{33}{2} (-2 + 30) = \frac{33}{2} \times 28 = 33 \times 14 = 462
$$

Now, multiply by $$4$$:

$$
4 \times 462 = 1,\!848
$$

### 2. Calculate $$\sum_{t=-2}^{30} 6$$

Since $$6$$ is a constant, the sum is simply:

$$
6 \times \text{Number of terms} = 6 \times 33 = 198
$$

### 3. Combine Both Results

Add the two sums together:

$$
1,\!848 + 198 = 2,\!046
$$

### **Final Answer**

$$
\sum_{t=-2}^{30} (4t + 6) = 2,\!046
$$
The sum of $$4t + 6$$ from $$t = -2$$ to $$30$$ is **2,046**.

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