If #k+1, 4k, 3k+5# is a geometric sequence
then the ratio between successive terms is equal.
#(k+1)/(4k) = (4k)/(3k+5)#
#rArr (k+1)(3k+5)=(4k)^2#
#rArr 3k^2+8k+5 = 16k^2#
#rArr 13k^2-8k-5=0#
We might be able to factor this directly or we could use the quadratic formula to determine the roots:
#color(white)("XXX")k= (8+-sqrt((-8)^2-4(13)(-5)))/(2(13)#
#color(white)("XXXX")= (8+-sqrt(324))/(2(13))#
#color(white)("XXXX")= (8+-sqrt(324))/(2(13)#
#color(white)("XXX")= (8+-18)/(2(13))#
#color(white)("XXXX")=(4+-9)/13#
#color(white)("XXXX")=13/13 = 1# or # = -5/13#
We could (and probably should) verify these results by checking that for each of these values of #k# the given sequence is geometric.
If #k=1#
then #k+1, 4k, 3k+5#
becomes #2, 4, 8# with an obvious common ratio of #2#
If #k=-5/13#
then #k+1, 4k, 3k+5#
becomes (with a little more effort) #8/13, -20/13, 50/13#
with a common ratio of #(-5/2)#
