# How do you find the first term and common difference that has a sum of its first 10 terms equal to 250 and whose 6th term is 32?

##### 1 Answer

The first term is

The common difference is

#### Explanation:

So, we are talking about arithmetic progression, that is (this is the definition) *a sequence of numbers, starting with some first one, #a#, and each consecutive one differed from the previous by the common difference #d#*.

That is, we are talking about a sequence

Assuming you don't remember the formula for this, let's derive a formula for a sum of the first

Since the sum does not change if we change the order of summation, we can summarize it in opposite order:

Sum both

Notice, that the results of summation in each

Therefore,

and

The problem states that for

Therefore, we have one equation:

(Eq. 1)

Since

(Eq. 2)

We have a system of two equations, Eq. 1 and Eq. 2, with two unknowns

It is easy to solve it using a substitution method.

From equation Eq. 2:

Substitute this value for

or

from which:

That is the common difference of our sequence.

Back to the unknown

*Checking* (**ALWAYS RECOMMENDED**)

The first 10 members of this sequence that starts with

Their sum is indeed

Checking confirms the validity of the answer.