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. 21​f(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+5​2​]+25−35​​(21−5​​)n[21−5​​−1−5​2​]
∴f(n+1)−f(n−1)=25+35​​(21+5​​)n(1)+25−35​​(21−5​​)n(1)=f(n)