Problem:
Find the eighth term of the sequence 1440,1716,1848,…, whose terms are formed by multiplying the corresponding terms of two arithmetic sequences.
Solution:
The nth term of an arithmetic sequence has the form an​=pn+q, so the product of corresponding terms of two arithmetic sequences is a quadratic expression, sn​=an2+bn+c. Letting n=0,1, and 2 produces the equations c=1440, a+b+c=1716, and 4a+2b+c=1848, whose common solution is a=−72, b=348, and c=1440. Thus the eighth term is s7​=−72⋅72+348⋅7+1440=348​. Note that sn​=−72n2+348n+1440=−12(2n−15)(3n+8) can be used to generate pairs of arithmetic sequences with the desired products, such as {180,156,132,…} and {8,11,14,…}.
The problems on this page are the property of the MAA's American Mathematics Competitions