Whats the missing term in the sequence 26. 16, __, 4?

If the last term was supposed to be #-4# then this could be an arithmetic sequence with common difference #-10# and the missing term is #6#

If the last term is actually #4# then this is neither an arithmetic sequence nor a geometric sequence, but could be a quadratic sequence with missing term #26/3#.

Actually I favour the idea that it's the sum of the arithmetic sequence:

#-1#, #-2#, #-3#, #-4#

and the geometric sequence:

#27#, #18#, #12#, #8#

which would make the third term #9#

I found this by looking at the approximate ratios between the terms, combined with some trial and error, but with hindsight I could have looked at it this way:

If #16#, _, #4# were a geometric sequence, then the middle term would be the geometric mean #sqrt(16*4)=8#. In this case the preceding term would have been #32#.

If #16#, _, #4# were an arithmetic sequence then the middle term would be the arithmetic mean #(16+4)/2=10#. In this case the preceding term would have been #22#.

Since the preceding term lies between these two values, the missing term lies somewhere between #8# and #10#. The only integer between them is #9#. So the remaining question is whether #9# works.

Write out the sequence:

#26, 16, 9, 4#

Write out the sequence of differences:

#-10, -7, -5#

Write out the sequence of differences of those differences:

#3,2#

Having taken differences twice we have eliminated any linear relationship - that is the constituent arithmetic sequence and are left with two terms whose ratio is that of the geometric sequence.

Since we're wanting to work with integer values, choose the first term of our geometric sequence as #27# to get:

#27,18,12,8#

Then subtract this from the original sequence to get:

#-1, -2, -3, -4#

Note that there are not enough constraints to determine a unique arithmetic and geometric sequence that can be added to give a sequence with first, second and fourth terms as given.

For example, if the arithmetic sequence is constant, then:

#a_n = ar^(n-1)+c#

with #a=(75+5sqrt(145))/4#, #r=(sqrt(145)-5)/10#, #c = (29-5sqrt(145))/4#

and the third term is #21-sqrt(145) ~~ 8.9584#