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?
The first term is
The common difference is
So, we are talking about arithmetic progression, that is (this is the definition) a sequence of numbers, starting with some first one,
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:
Notice, that the results of summation in each
The problem states that for
Therefore, we have one equation:
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
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.