Question

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?

Answer 1

The first term is `-38` (negative `38`).
The common difference is `14`.

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
`a, a+d, a+2*d, a+3*d,..., a+N*d,...`

Assuming you don't remember the formula for this, let's derive a formula for a sum of the first `N` terms of this sequence:
`S_N = [a] + [a+d] + [a+2*d] +...+ [a+(N-2)*d] + [a+(N-1)*d]`

Since the sum does not change if we change the order of summation, we can summarize it in opposite order:
`S_N = [a+(N-1)*d] + [a+(N-2)*d] +...+ [a+2*d] + [a+d] + [a]`

Sum both `S_N` adding first term to first term, second term to second term etc., `(N-1)^(th)` term to corresponding `(N-1)^(th)` term, getting:

`S_N+S_N =`
`= {[a] + [a+(N-1)d]} +`
`+ {[a+d] + [a+(N-2)d]} +...`
`...+ {[a+(N-2)d] + [a+d]} +`
`+ {[a+(N-1)d] + [a]}`

Notice, that the results of summation in each `{...}` is the same, `2a+(N-1)*d`.

Therefore,
`2S_N= {2a+(N-1)*d}*N`

and
`S_N = a*N + [(N-1)*N*d]/2`

The problem states that for `N=10` `S_10=250`.
Therefore, we have one equation:
(Eq. 1) `250 = 10a+45d`

Since `6^(th)` term is `32`, we have another equation:
`a+(6-1)d=32` or
(Eq. 2) `a+5d=32`

We have a system of two equations, Eq. 1 and Eq. 2, with two unknowns `a` and `d`.
It is easy to solve it using a substitution method.

From equation Eq. 2:
`a = 32-5d`

Substitute this value for `a` into equation Eq. 1:
`250 = 10(32-5d)+45d`
or
`250 = 320-50d+45d`,
from which:
`5d=70` and
`d=14`
That is the common difference of our sequence.

Back to the unknown `a=32-5d`:
`a=32-5*14=-38`

Checking (ALWAYS RECOMMENDED)

The first 10 members of this sequence that starts with `-38` and adds `14` to each new member are
`-38, -24, -10, 4, 18, 32, 46, 60, 74, 88`.
Their sum is indeed `250` and their `6^(th)` term is indeed `32`.
Checking confirms the validity of the answer.