you are given the terms 1-24x+252x2 and also the bionomial from which they were expanded... (1+ax)n
By examining the second term...
T2=nC1(1)n-1(ax)1
but from the exapnsion we know the second term is -24x
.: -24x = nC1(1)n-1(ax)1 {nC1=n}
.: by eliminating we get -24=na [1]
also...
T3=nC2(1)n-2(ax)2
but T3=252x2
.: 252x2=nC2(1)n-2(ax)2
252x2= (n!/[2!(n-2)!]).1.a2x2 {the factorial can be simplified into n(n-1)/2}
252=a2n(n-1)/2
.: 504=a2n2 - a2n [2]
by substituting [1] in....
-72 = -a2n [3]
Now... by dividing [3] by [1] we get...
a=-3 also, by substituting a back into [1] we get n=8
Therefore, these terms were created by the expansion of the binomial (1-3x)8