Problem: Let f(n)=105+35(21+5)n+105−35(21−5)n. Then f(n+1)−f(n−1), expressed in terms of f(n), equals:
Answer Choices:
A. 21f(n)
B. f(n)
C. 2f(n)+1
D. f2(n)
E. 21(f2(n)−1)
Solution:
f(n+1)−f(n−1)=25+35(21+5)n+1+25−35(21−5)n+1−25+35(21+5)n−1−25−35(21−5)n−1
∴f(n+1)−f(n−1)=25+35(21+5)n[21+5−1+52]+25−35(21−5)n[21−5−1−52]
∴f(n+1)−f(n−1)=25+35(21+5)n(1)+25−35(21−5)n(1)=f(n)