# Find an equation for the nth term of the arithmetic sequence.

a_{10} = 32, a_{12} = 106

**Solution:**

The nth for the arithmetic sequence is given by:

a_{n} = a + (n - 1)d --- (1)

Where, a is the first term

d is the common difference of two consecutive terms.

n is the number of terms.

It is given that:

a_{10} = 32 and a_{12} = 105

From equation 1 we have,

⇒ a_{10} = a + 9d

⇒ a + 9d = 32 --- (2)

⇒ a_{12} = a + 11d

⇒ a + 11d = 106 --- (3)

Subtract equation [2] from [3] we have;

⇒ a + 11d - a - 9d = 106 - 32

By simplification we get,

⇒ 2d = 74

By dividing both sides by 2 we get,

⇒ d = 37

Substitute the value of d in [2] we have;

⇒ a + 9(37) = 32

⇒ a + 333 = 32

Subtract 333 from both sides we have;

⇒ a = -301

Then substitute the value a and d in equation [1] we have;

⇒ a_{n} = - 301 + (n - 1)(37)

⇒ a_{n} = - 301 + 37n - 37

⇒ a_{n} = - 338 + 37n

Therefore, a_{n} = - 338 + 37n an equation for the nth term.

## Find an equation for the nth term of the arithmetic sequence.

a_{10} = 32, a_{12} = 106

**Summary:**

a_{n} = - 338 + 37n an equation for the nth term of the arithmetic sequence. a_{10} = 32, a_{12} = 106.

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