A sequence of four integers is both arithmetic and geometric. if the sum of the four numbers is 20, find the first integer in the sequence.

Answer 1

5. The only sequences that are both
arithmetic and geometric are constant
sequences, in which all the terms must
be the same. For us to have a sum of 20,
each term of the sequence must be 5. We
can show this algebraically by solving
the system of equations a + d = ar and
a + 2d = ar² where a represents the first term of the sequence, d the common
difference of the arithmetic sequence,
and r the common ratio of the geometric sequence. This system produces the
solution d = 0 and r = 1.