Given ________, – 243, – 226, – 209, – 192, – 175, …

a. What is the 67th term of the sequence
b. What is the series of the first 27 terms of the sequence

a = -243

d = -226 - (-243) = 17
term(67) = a + 66d
= -243 + 66(17) = 879

sum(n) = (n/2)(2a + (n-1)d)
sum (27) = (27/2)(-486 + 26(17))
= -594

Here is a nice new part to the question:
For which terms does the sequence switch from negative to positive ?

Just looked at the first of the "Related questions" below, which I had answered in June, and realized that your _____ , meant that -243 was actually the second term

I am pretty sure you can make the suitable adjustments.
(your first term will be -260)

To determine the 67th term of the sequence, we need to find the pattern in the given sequence. In this case, the given sequence seems to be an arithmetic sequence because there is a common difference between each term.

a. To find the 67th term of the sequence, we need to determine the formula for the arithmetic sequence. Let's start by identifying the common difference:

The common difference can be calculated by subtracting any two consecutive terms. Taking the difference between -243 and -226, we get:

-243 - (-226) = -243 + 226 = -17

So, the common difference is -17.

The formula for the nth term of an arithmetic sequence is given by:

an = a1 + (n - 1)d

where:
an is the nth term,
a1 is the first term,
n is the number of terms,
d is the common difference.

In our case, the first term is -243 and the common difference is -17. Plugging these values into the formula, we get:

a67 = -243 + (67 - 1)(-17)

a67 = -243 + 66(-17)

a67 = -243 - 1122

a67 = -1365

Therefore, the 67th term of the sequence is -1365.

b. To find the series of the first 27 terms of the sequence, we can apply the formula for the nth term as we did in part a.

The formula for the nth term of an arithmetic sequence is:

an = a1 + (n - 1)d

In this case, the first term is -243 and the common difference is -17. Plugging these values into the formula, we can find the first 27 terms:

a1 = -243
d = -17

To find the first 27 terms, substitute the values of n from 1 to 27 into the formula:

a1 + (1 - 1)(-17) = -243
a1 + (2 - 1)(-17) = -226
a1 + (3 - 1)(-17) = -209
a1 + (4 - 1)(-17) = -192

Continuing this pattern, we can find all 27 terms of the sequence.