Inquiry Starting Point The sum, difference, product and quotient of two numbers form the first four numbers of an arithmetic sequence. Is there a fifth term? Does this have a solution? Solution Let the two numbers be x and y. As a startig point, and without loss of generality, suppose that the sum is the first term, the difference the second, the product the third term and so on. Then the sequence structure has: First term a := x + y Second term = x − y Third term = xy Fourth term = x/y. As this sequence is arithmetic, define the difference between terms as d. At this point, note that not we have not restricted the values of x, y, a, d to be integers, or indeed, real. Thus, the difference can be found from second term − first term
d =
(x − y) − (x + y) = −2y
=
(1)
third term − second term
d =
= xy − (x − y) = xy − x + y
(2)
fourth term − third term x − xy = y
(3)
and d =
Setting (1) = (2) gives xy − x + y
= −2y
x(y − 1)
= −3y 3y = 1−y
x
(4)
Similarly, to set (2) = (3) gives xy − x + y xy 2 − xy + y 2 2
x − xy y = x − xy 2 =
2xy − xy + y 2 + x
=
0
2
x(2y − y + 1)
=
−y 2
x
=
y2 y − 1 − 2y 2
(5)
We now have two equations (4) and (5) of x given in terms of y. Thus, setting (4) equal to (5) gives y2 y − 1 − 2y 2 so y 2 − y 3
=
3y 1−y 3y 2 − 3y − 6y 3
5y 2 − 2y + 3
=
0
so y
=
=
and dividing by y gives
taking the positive discriminant gives x =
1
√ 0.2 ± 0.2i 14
√ 2.65 + 0.75i 14
(6)
A second pair of x− and y− would also be generated. Using (x+ , y+ ) as a solution, the first term of the sequence is a+
= x+ + y+ = =
√ √ 0.2 + 0.2i 14 + 2.65 + 0.75i 14 √ 2.85 + 0.95i 14
and, thus, from (1) d+
= −2y+
√ = −2(0.2 + 0.2i 14) √ = −0.4 − 0.4i 14
This sequence has been defined from the (x+ , y+ ) values. A different a− and d− are given rise to from the (x− , y− ) solution. So, what is the fifth term? a5
= a + 4d = =
√ √ 2.85 + 0.95i 14 + 4(−0.4 − 0.4i 14) √ 1.25 − 0.65i 14
Next Steps... This is one solution for the arithmetic progresssion, based on the assumption that the first term is the sum, the second the product etc, from the inquiry prompt. If you were to mix the order of those definitions, you could have the first term the sum, the second the product, the third the difference, the fourth the quotient etc. If you were to fully investigate all combinations, this leads to up to 4! = 24 possibilities. The story doesn’t end there, however. It is up to 4! possibilities, assuming that one rearrangement may yield 1 solution for x and y. As we saw in this case, there are two solutions (we only applied one of the solution pairs for x = x+ and y = y+ ). If other permutations of the assumption of the terms were to produce a cubic, this could yield 3 cases, a quartic, 4 cases etc. The up to 24 possibilities clause comes with the caviat that the algebraic relationships generated from the different permutations on the definition could give rise to many more. So how many more? The quadratic denominator term on line (5) was produced due to the denominator in (3) coming from the quotient in the enquiry prompt. This lead to the presence of a cubic on line (6). We can assume that combinations of different starting points would not generate a higher power than 4, as there may quadratics on both numerators and denominators of any relationship defining d, cross products of which would produce quartic expressions. As we will not find a different situation producing higher than a power of 4, it is given that at most we will have 4 × 4! = 96 possible solutions. If you restric this to only real cases, then two solutions in the quadratic and four solutions in the quartic cases may be lost. To look at the minumum number of cases that this would produce, only linear and cubic cases would generate real solutions. The linear case could produce no solution, (in this case for example: 2(x + 4) = 2x + 7, were it to arise - a mathematical contradiction that could come from y = 2(x + 4) and y = 2x + 7)) and, similarly, with a cubic, you may end up with no solution. To restrict this problem to the real domain may give no solutions... Thus, my enquiry concludes with the statement that this problem has between 0 and 96 distinct solutions, of which one is cited here.
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