First member and common difference of arithmetic sequence given the following two equations

Question

So I'm completely stumped with this one. Tried tackling it from a few different angles but getting nowhere. I have to figure out $a_1$ and the common difference of the sequence where the following two equation hold true.

$a_2+a_3+a_4=20\\ a_1*a_4=-20 $

Answer 1

HINT Let $d$ be the difference. Then, $a_2 = a_1+d$ and $a_3 = a_1+2d$. Can you write $a_4$ as a function of $a_1$ and $d$ and plug into both equations?

UPDATE

Correct, $a_4 = a_1+3d$ so the first equation becomes $$ 20 = a_2 + a_3 + a_4 = 3a_1 + 6d $$ and the second one is $$ a_1(a_1+3d) = -20 $$ and you can use the first equation to solve for $3d$ and plug into the second one: $3d=10-3a_1/2$, thus we get $$ a_1(a_1+10-3a_1/2) = -20 $$ Can you finish?

Answer 2

This can also be solved using the sum of an arithmetic sequence in terms of its first and last terms, $ \ s \ = \ \frac{n}{2}·(a_1 + a_{n}) \ \ . $ The first equation indicates a sequence of four terms, for which the given sum is missing the first term. So we can write $$ s \ \ = \ \ \frac{4}{2}·(a_1 + a_{4}) \ \ = \ \ a_1 \ + \ a_2 \ + \ a_3 \ + \ a_4 \ = \ 20 \ + \ a_1 $$ $$ \Rightarrow \ \ 2·a_1 \ + \ 2·a_4 \ \ = \ \ 20 \ + \ a_1 \ \ \Rightarrow \ \ a_4 \ \ = \ \ 10 \ - \ \frac{a_1}{2} \ \ . $$ This can then be inserted into the second equation to produce $$ a_1 · a_4 \ \ = \ \ a_1 · \left(10 \ - \ \frac{a_1}{2} \right) \ \ = \ \ -20 \ \ \Rightarrow \ \ a_1^2 \ - \ 20·a_1 \ - \ 40 \ \ = \ \ 0 $$ $$ \Rightarrow \ \ a_1 \ \ = \ 10 \ \pm \ 2\sqrt{35} \ \ \Rightarrow \ \ a_4 \ \ = \ \ 10 \ - \ \frac{10 \ \pm \ 2\sqrt{35}}{2} \ \ = \ \ 5 \ \mp \ \sqrt{35} \ \ . $$ Since $ \ a_4 \ = \ a_1 \ + \ 3d \ \ , $ we then conclude that $$ \ d \ = \ \frac{(5 \ \mp \ \sqrt{35}) \ - \ (10 \ \pm \ 2\sqrt{35})}{3} \ \ = \ \ \frac{-5 \ \mp \ 3\sqrt{35}}{3} \ \ = \ \ -\frac53 \ \mp \ \sqrt{35} \ \ . $$ We have two results for the first term and the difference between terms -- one positive, one negative -- suggesting that there are two arithmetic sequences that meet the stated conditions:

$$ 10 \ - \ 2\sqrt{35} \ \ , \ \ \frac{25}{3} \ - \ \sqrt{35} \ \ , \ \ \frac{20}{3} \ \ , \ \ 5 \ + \ \sqrt{35} $$ $$ ( \ \approx \ \ -1.832 \ \ \ , \ \ \ 2.417 \ \ \ , \ \ \ 6.667 \ \ \ , \ \ \ 10.916 \ ) $$ [the increasing sequence]

and $$ 10 \ + \ 2\sqrt{35} \ \ , \ \ \frac{25}{3} \ + \ \sqrt{35} \ \ , \ \ \frac{20}{3} \ \ , \ \ 5 \ - \ \sqrt{35} $$ $$ ( \ \approx \ \ 21.832 \ \ \ , \ \ \ 14.249 \ \ \ , \ \ \ 6.667 \ \ \ , \ \ \ -0.916 \ ) $$ [the decreasing sequence].

Both of these arithmetic progressions do indeed satisfy the specified conditions.