The first term of a linear sequence is 7.5 and the fourth term is 3.find the number of terms in the sequence if its sum is -9.

use your definitions.

a = 7.5
a+ 3d = 3
7.5 + 3d = 3
d = -1.5

sum(n) = (n/2)(2a + (n-1)d)
-9 = (n/2)(15 + (n-1)(-1.5) )
-18 = n(15 - 1.5n + 1.5)
-18 = 16.5n - 1.5n^2
double and re-arrange
3n^2 - 33n - 36 = 0
n^2 - 11n - 12 = 0
(n+1)(n-12) = 0
n = -1 or n = 12

there are 12 terms

To find the number of terms in the sequence, we need to determine the common difference first.

The formula for the nth term of a linear sequence is given by:

\[T_n = a + (n - 1)d\]

where \(T_n\) is the nth term, \(a\) is the first term, \(n\) is the number of terms, and \(d\) is the common difference.

Given that the first term (\(a\)) is 7.5 and the fourth term is 3, we can write two equations using the formula above:

1. \(7.5 + 3d = 3\)
2. \(7.5 + (n - 1)d = -9\)

Solving equation 1 for \(d\):
\[3d = 3 - 7.5\]
\[3d = -4.5\]
\[d = -1.5\]

Now, we substitute the value of \(d\) into equation 2:
\[7.5 + (n - 1)(-1.5) = -9\]

Simplifying the equation:
\[7.5 - 1.5n + 1.5 = -9\]
\[9 + 7.5 = -1.5n\]
\[16.5 = -1.5n\]

Divide both sides by -1.5 to solve for \(n\):
\[n = \frac{16.5}{-1.5}\]
\[n = -11\]

The number of terms in the sequence is -11. However, it is not possible to have a negative number of terms in a sequence. Therefore, we can conclude that there must be an error in the given information or question.

To find the number of terms in the sequence, we need to determine the common difference between terms.

The formula for the nth term of a linear sequence is:

aₙ = a₁ + (n - 1)d

Where:
aₙ is the nth term,
a₁ is the first term,
n is the position of the term,
d is the common difference.

Given that the first term (a₁) is 7.5 and the fourth term (a₄) is 3, we can substitute these values into the formula:

7.5 + (4 - 1)d = 3

Simplifying the equation:

7.5 + 3d = 3

Subtracting 7.5 from both sides:

3d = 3 - 7.5
3d = -4.5

Dividing both sides by 3:

d = -4.5 / 3
d = -1.5

Now that we have the common difference (d), we can find the number of terms (n) using the formula for the sum of a linear sequence:

Sₙ = (n / 2)(2a₁ + (n - 1)d)

Given that the sum (Sₙ) is -9, we can substitute the known values:

-9 = (n / 2)(2(7.5) + (n - 1)(-1.5))

Simplifying further:

-9 = (n / 2)(15 + (-1.5n + 1.5))

-9 = (n / 2)(15 - 1.5n + 1.5)

-9 = (n / 2)(16.5 - 1.5n)

Multiplying both sides by 2 to eliminate the fraction:

-18 = n(16.5 - 1.5n)

Expanding and rearranging the equation:

-18 = 16.5n - 1.5n²

1.5n² - 16.5n - 18 = 0

To solve this quadratic equation, we can either factorize or use the quadratic formula. Let's use the quadratic formula:

n = (-b ± sqrt(b² - 4ac)) / (2a)

For our equation, the values of a, b, and c are:

a = 1.5
b = -16.5
c = -18

Substituting the values:

n = (-(-16.5) ± sqrt((-16.5)² - 4(1.5)(-18))) / (2(1.5))

Simplifying:

n = (16.5 ± sqrt(272.25 + 108)) / 3

n = (16.5 ± sqrt(380.25)) / 3

n = (16.5 ± 19.5) / 3

We get two possible solutions:
n₁ = (16.5 + 19.5) / 3
n₂ = (16.5 - 19.5) / 3

Calculating:

n₁ = 36 / 3 = 12
n₂ = -3 / 3 = -1

Since the number of terms (n) cannot be negative, we discard n₂ = -1 as an extraneous solution.

Therefore, the number of terms in the sequence is n = 12.