Problem:
In the sequence
…,a,b,c,d,0,1,1,2,3,5,8,…
each term is the sum of the two terms to its left. Find a.
Answer Choices:
A. −3
B. −1
C. 0
D. 1
E. 3
Solution:
Calculate d,c,b and a in that order from the definition of the sequence: 1=d+0, so d=1; then 0=c+d, so c=−1; next 1=d=b+c=b−1, so b=2; finally, −1=c=a+b=a+2, so a=−3.
OR
Use the rule for generating terms of the sequence beginning with the two terms a,b:
a,b,a+b,a+2b,2a+3b,3a+5b,…
to see that 2a+3b=0 and 3a+5b=1. Solve simultaneously to find a=−3.