Indicate the general rule for the arithmetic sequence with a₃ = -12 and a₈ = -37.

Solution:

A sequence where the difference between each successive pair of terms is the same is called an arithmetic sequence.

The general rule of arithmetic sequence is

a_n = a₁ + (n - 1)d

Where a₁ - first term

d - common difference

n - number of terms

It is given that

a₃ = -12 and

a₈ = -37

If a₃ = -12 using the arithmetic sequence rule we get

a + (3 - 1) d = -12

a + 2d = -12

We can write it as

a = - 12 - 2d …. (1)

If a₈ = -37 using the arithmetic sequence rule we get

a + (8 - 1) d = -37

a + 7d = -37 ….. (2)

Let us substitute equation (1) in (2)

- 12 - 2d + 7d = -37

- 12 + 5d = -37

5d = - 37 + 12

5d = -25

Dividing both sides by 5

d = - 5

Substituting the value of d in equation (1)

a = -12 - 2 (-5)

a = -12 + 10

a = - 2

Therefore, the general rule for the arithmetic sequence is an = - 2 + (n - 1) (-5).

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Indicate the general rule for the arithmetic sequence with a₃ = -12 and a₈ = -37.

Summary:

The general rule for the arithmetic sequence with a₃ = -12 and a₈ = -37 is an = - 2 + (n - 1) (-5).