1.6 The sum of the first 15 terms of an arithmetic progression is 165 . The common difference is 2 . Calculate the first term of the sequence.

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To find the first term of the arithmetic progression (AP), we can use the formula for the sum of the first $$ n $$ terms of an AP:

$$
S_n = \frac{n}{2} \times (2a + (n-1)d)
$$

where:
- $$ S_n $$ is the sum of the first $$ n $$ terms,
- $$ n $$ is the number of terms,
- $$ a $$ is the first term,
- $$ d $$ is the common difference.

Given:
- $$ S_{15} = 165 $$
- $$ n = 15 $$
- $$ d = 2 $$

We can substitute these values into the formula:

$$
165 = \frac{15}{2} \times (2a + (15-1) \cdot 2)
$$

Simplifying the equation:

$$
165 = \frac{15}{2} \times (2a + 28)
$$

Now, multiply both sides by 2 to eliminate the fraction:

$$
330 = 15 \times (2a + 28)
$$

Next, divide both sides by 15:

$$
22 = 2a + 28
$$

Now, isolate $$ 2a $$:

$$
2a = 22 - 28
$$
$$
2a = -6
$$

Now, divide by 2 to find $$ a $$:

$$
a = -3
$$

Thus, the first term of the sequence is:

$$
\boxed{-3}
$$
The first term of the sequence is -3.

1.6 The sum of the first 15 terms of an arithmetic progression is 165 . The common difference is 2 . Calculate the first term of the sequence.

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