Question

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

a10 = 32, a12 = 106

Answer 1

an= -301+37(n-1) is the answer

Answer 2

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.

As per the statement:

`a_(10) = 32` and
`a_(12) =106`

Using [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`

Simplify:

`2d = 74`

Divide 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 [1] we have;

`a_n=-301+(n-1)(37)`

⇒
`a_n = -301+37n-37 = -338+37n`

Therefore, an equation for the nth term of the arithmetic sequence is :

`a_n = -338+37n`