Question

Let d and p be two real numbers.

Answer 1

1 Find the first term of an arithmetic
progression \[\huge{a_1,a_2,a_3...}\]
with difference d such that
\[\huge{\color{red }{a_1a_2a_3a_4=p}}\]
2 Find the number of solutions in terms of d and p.

Answer 2

\[\huge \color{brown}{ MEDALS \lceil AVAILABLE \rceil }\]

Answer 3

i am just trying...
The product of ‘n’ terms of a finite AP is
\(\large d^n \frac{\Gamma(a_1/d+n)}{\Gamma(a_1/d)}\)
here, it'll be
\(\large p= d^4 \frac{\Gamma(a_1/d+4)}{\Gamma(a_1/d)} \\ p=d^4(a_1/d+4)(a_1/d+3)(a_1/d+2)(a_1/d+1) \\ [as, \Gamma (n)=n(\Gamma (n-1))]\)
\(p=(a_1+4d)(a_1+3d)(a_1+2d)(a_1+d)\)
need to find a1 !....got overcomplicated.
(and by the way, that you have got just by using n'th term formula :P)
maybe you can get some ideas how to continue from here....
since this is a good question, i would like to tag some people, if they don't mind, who i think can solve this, maybe..
@jim_thompson5910 @joemath314159 @Zarkon @mahmit2012

Answer 4

\[a_n=a_1+(n-1)d\]
\[S_4=\dfrac{4(2a_1+3d)}{2}\]
\[S_4=4a_1+6d\]

Answer 5

thanks @hartnn

Answer 6

@mukushla